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Active Neural Matter: A Finite-Window Response Framework for Physical Systems and Distributed AI Workflows
Tianchi Chen, Shihua Yu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 10/14/2026, 10:09:41 AM
Summary
The paper introduces Active Neural Matter (ANM), a finite-window response framework for modeling physical systems and distributed AI workflows. ANM addresses the problem of determining what past information a system's state must retain to predict future responses. It employs forward analysis to predict state and output responses to perturbations, separating propagation from output sensitivity, and inverse analysis to test if a candidate state retains sufficient information. The framework uses uncertainty bounds and delayed-consensus simulations to validate analytical predictions and establish sufficient states for specified programs or tasks.
Entities (10)
Relation Signals (7)
Tianchi Chen → affiliatedwith → Merck & Co., Inc.
confidence 95% · 1Merck & Co., Inc., Rahway, NJ, USA
Shihua Yu → affiliatedwith → Northeastern University
confidence 95% · 2Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA
Active Neural Matter → uses → Finite-Window Response Framework
confidence 95% · Active Neural Matter (ANM) addresses this problem through a finite-window response framework.
Active Neural Matter → appliesto → Distributed AI Workflows
confidence 92% · Physical systems and distributed AI workflows can retain past information... Active Neural Matter (ANM) addresses this problem
Active Neural Matter → appliesto → Physical Systems
confidence 92% · Physical systems and distributed AI workflows can retain past information... Active Neural Matter (ANM) addresses this problem
Active Neural Matter → employs → Inverse Analysis
confidence 90% · Inverse analysis tests whether a candidate state retains enough information for these responses.
Active Neural Matter → employs → Forward Analysis
confidence 90% · Forward analysis uses each system's dynamics to predict state and output responses to perturbations
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Abstract
Abstract:Physical systems and distributed AI workflows can retain past information, receive delayed feedback and combine conflicting signals. These features pose a common modeling problem: determining what information about past interactions a state must retain to account for future responses. Active Neural Matter (ANM) addresses this problem through a finite-window response framework. Forward analysis uses each system's dynamics to predict state and output responses to perturbations, separating propagation from output sensitivity. Inverse analysis tests whether a candidate state retains enough information for these responses. With the system and future inputs or interaction rules fixed, distinguishable responses from histories assigned the same state identify distinctions the representation must preserve. Their distributional separation gives a lower bound on the worst-case error of approximating behavior from that state. Uncertainty bounds distinguish rejection, support within a fixed tolerance on tested comparisons, and unresolved cases. Delayed-consensus simulations check analytical predictions of response amplitude and phase over a narrow range of delays; flocking illustrates dependence on the chosen observable. An exact delayed-return construction establishes a sufficient state for a specified program. The AI studies identify distinctions in interaction history and evidence order that a representation must preserve to reproduce observed response distributions. Tests on new histories in a controlled feedback task provide bounded support for a state specified by the task instructions; the instructed rule predicts every observed action. ANM connects response prediction to measurable information requirements, separating behavioral fidelity from task correctness and initial-state adequacy from a closed evolution law for that state.
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- Source: https://arxiv.org/abs/2610.14549v1
- Canonical: https://arxiv.org/abs/2610.14549v1
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Active Neural Matter: A Finite-Window Response Framework for Physical Systems and Distributed AI Workflows Tianchi Chen1,*, Shihua Yu2 1Merck & Co., Inc., Rahway, NJ, USA 2Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA *Corresponding author: chentianchi@gmail.com Abstract Physical systems and distributed AI workflows can retain past information, receive delayed feedback and combine conflicting signals. These features pose a common modeling problem: determining what information about past interactions a state must retain to account for future responses. Active Neural Matter (ANM) addresses this problem through a finite-window response framework. Forward analysis uses each system’s dynamics to predict state and output responses to perturbations, separating propagation from output sensitivity. Inverse analysis tests whether a candidate state retains enough information for these responses. With the system and future inputs or interaction rules fixed, distinguishable responses from histories assigned the same state identify distinctions the representation must preserve. Their distributional separation gives a lower bound on the worst-case error of approximating behavior from that state. Uncertainty bounds distinguish rejection, support within a fixed tolerance on tested comparisons, and unresolved cases. Delayed-consensus simulations check analytical predictions of response amplitude and phase over a narrow range of delays; flocking illustrates dependence on the chosen observable. An exact delayed-return construction establishes a sufficient state for a specified program. The AI studies identify distinctions in interaction history and evidence order that a representation must preserve to reproduce observed response distributions. Tests on new histories in a controlled feedback task provide bounded support for a state specified by the task instructions; the instructed rule predicts every observed action. ANM connects response prediction to measurable information requirements, separating behavioral fidelity from task correctness and initial-state adequacy from a closed evolution law for that state. 1 Introduction A local perturbation can remain in a system after the interaction that produced it has ended. Its later effect depends on how the system carries that change forward and how the chosen observable responds. In a delayed physical network, two identical instantaneous fields can have different futures because their recent histories differ. In an AI workflow, an identical tool result can have different meanings because it answers a different earlier request. These examples pose a common question: what information must a state description preserve to account for a specified response over a finite observation window? A candidate description of information carried forward by a system is its retained state. It may be a physical field with its recent history, a set of communication records, or selected coordinates of either. A readout is a measured observable, or a vector of observables; an endpoint readout is evaluated at the end of the chosen window. A readout can also summarize that window, provided the necessary observations are retained. These definitions separate the state being proposed from the measurements used to test it. Active Neural Matter (ANM) treats a controlled input change as a source perturbation and measures its effects on the state and readout separately over a finite window (Figure 1). The term neural is inspired by neural dynamics, where past inputs can influence later responses through persistent activity 14 or activity-dependent changes in synaptic efficacy (synaptic plasticity) 34. ANM abstracts this relationship into source perturbations, retained states and readouts. Information may be retained in field histories, internal memory or workflow records; each realization supplies its own retention mechanism and dynamics. The choice of representation becomes an experimental hypothesis. Forward analysis predicts responses from specified states and dynamics. Inverse analysis tests whether a candidate state retains enough information: with the system and continuation conditions fixed, a resolved readout difference between identical candidate states reveals an omitted distinction. Stochastic outputs require distributional or statistical comparisons; approximate state matches also require sensitivity and measurement-error bounds (Section 2). These tests constrain adequacy without uniquely identifying a missing variable. The description applies to continuous physical dynamics and discrete, event-driven workflows, with realization-specific states, dynamics and response functions. Here distributed AI workflow means that producing, retaining and using information occur in separate components. A downstream component reads the records and produces an output; it can be a deterministic program or a language model. The communication records permit response measurements without access to model weights. An externally specified graph and the request–return example in Section 5 instantiate these roles; neither requires or identifies a state jointly formed by interacting live agents. This organization draws on response theory 23, 24, 46, projection methods with memory 37, 69, and state-based diagnosis 16. Section 9 compares its measurement access and inferential targets with these foundations and with workflow, memory and intervention methods. The contributions comprise a common response formulation and its forward and inverse uses: 1. A common response formulation across physics and AI. Source perturbations, retained states and readouts define a common finite-window response problem, with dynamics supplied by each realization. Keeping history and readout choice explicit connects perturbation propagation to the information needed for the resulting observations. 2. Predict and separate responses. Paired state and readout measurements distinguish propagation from readout gains in a linear model, as in standard state-space identification (Section 2). Finite-window predictions retain the history required by delays. Nested readout criteria distinguish continued operation from correct operation; other readouts need not be nested. 3. Test whether a representation retains sufficient information. Comparisons of response slopes or finite changes distinguish rejection, bounded support within a preset tolerance, and unresolved adequacy. Resolved readout differences at identical candidate states identify histories that an adequate representation must distinguish. Their separation also gives a lower bound on the worst-case behavioral approximation error of merging them (Section 8.3). New histories test a revision with the program or model and its interface unchanged. In a controlled closed-loop workflow, this procedure tests the version distinction prescribed by the instructions, rejecting version-omitting summaries and supporting a predefined refinement on new histories; the instructed rule predicts every observed action (Section 6). The physical and graph studies first compare predicted and measured responses. The inverse studies then ask what information must be retained when future inputs are fixed, when actions select later observations, and when a model judges scientific text. These comparisons lead to the information requirements and approximation-error bound in Section 8. The model results remain conditional on the tested histories and readouts; Sections 5.3–5.5 examine how support changes with the choice of recipient, identifiers and request order. Figure 1: Forward response and inverse representation tests. a, A finite source contrast Δu u produces separately measured candidate-state and observable changes ΔzT,ΔOT z_T, O_T. Solid arrows denote measurements; the dashed state-to-readout relation is a hypothesis, not an established causal path. b, Histories a,ba,b with identical states but deterministic readout differences beyond measurement error must be distinguished by an adequate representation. Stochastic readouts require resolved differences in the specified statistic or conditional distribution. The continuation fixes later inputs or the policy and environment rules generating them; approximate matches require sensitivity and error bounds (Section 2). Panel b describes states at the end of the observation window. A test of a preparation state instead compares initial summaries under the same continuation conditions with the readouts obtained afterward. 2 A common response framework for physical and AI systems The common structure connects a source change to both retained information and a measured observable. Continuous evolution and discrete transitions supply different response maps, while the same representation question applies: does the candidate state preserve every distinction needed for the specified readout? Differentiable responses admit a local linear test; finite changes are compared directly. Each realization supplies its state space, dynamics, units and noise law. In a computational graph, a typed source event specifies an update time, graph coordinate and evidence role; its numerical amplitude is the controlled source. Table 1 maps the common measurement roles to the physical and computational operations. To compare readouts across realizations, specify a correspondence between reached states. Readouts agree on corresponding states exactly when their maps agree under that correspondence (SI Section S1.5). The reported systems share measurement roles, without an assumed correspondence between their states. Table 1: The same measurement roles, with different dynamics. Source, retained information and readout are specified separately. The last column states the question answered by each realization; it does not assert equality of their response functions. Realization Controlled input Retained information and readout Scientific role Delayed consensus Applied drive; coupling delay or noise in separate sweeps Field and recent history; measured amplitude, phase and spatial variance Calibrate the predicted driven response against an analytic solution. Polar flock Angular-noise amplitude at fixed aligning field Positions and headings; polarization and angular alignment Compare nonlinear steady-state observables and their dependence on the chosen criterion. Typed-event graph Amplitude of an event with specified time, site and channel Numerical graph state; a program or model reads specified terminal coordinates Measure imposed propagation and readout response separately. Delayed-tool workflow Preparation histories or subsequent sensor returns Complete request–return records; analyst-selected descriptions and a fixed downstream component’s output Test which historical distinctions a candidate representation must preserve. Adaptive test workflow Which test belongs to an old version; new test outcome; return delivery Initial test records; distribution of model actions before and after a selected tool return Test state information and conditional predictions when actions determine later observations. Scientific-text comparison Excerpt selection or document order Selected abstract text; the reader’s conditional judgment distribution Test whether document summaries or unordered excerpts preserve the measured response distinctions. In the graph realization, an extractor forms events from text, prescribed updates transport their contributions, and a model reads the terminal scores (Section 4). This open-loop experiment separates propagation from model response without readout feedback or an identification of the graph update with the physical field equation. The adaptive test workflow instead couples model actions to tool observations: the chosen test determines the next request, and its delivered return affects the next decision (Section 6). Its action policy and environment rule define discrete stochastic dynamics; testing an initial summary does not assume autonomous reduced dynamics. The inverse studies use records directly: the downstream component receives complete histories or ordered excerpts, while candidate states are computed only for analysis. Refining a candidate leaves the component and each input unchanged. Controlled histories, excerpt selections or orderings provide the comparisons; editing the component or its input would test a system intervention. The full state sts_t contains the histories and records required for evolution and measurement. A candidate retained state zt=Gf(st)z_t=G_f(s_t) selects information from it; the tests ask whether this selection preserves the specified response. Here f fixes experimental conditions, and Gf:Xf→ZfG_f:X_f→ Z_f maps the full-state space to the candidate state space. A physical field ctc_t may be part of sts_t; with delay, its current value alone need not determine the future. Sufficiency is relative to the readout, window and continuation. Choosing a representation for analysis leaves the system and its inputs unchanged. A delayed return makes this distinction concrete. Two histories have identical snapshots—no return and two pending requests—but exchange which sensor was requested for recipients A and B. Under the same later returns, 2 from sensor 1 and 7 from sensor 2, a program matching returns to requests reports 2 for A in one history and 7 in the other. The snapshot merges histories that the response distinguishes; retaining the association separates them. Section 5 establishes exact sufficiency for this fixed program and then tests whether the association state also preserves a language model’s responses. 2.1 Fixed inputs and adaptive observations A preparation-state test compares histories before the measured evolution. Let h denote that history and z=Gf(h)z=G_f(h) its candidate summary. In an open-loop experiment the later input sequence is specified externally. In a closed-loop workflow an action changes the next observation, so fixing that sequence would suppress the adaptation being tested. A discrete description makes this distinction explicit. Let hth_t be the visible history before decision t, ata_t the selected action and rt+1r_t+1 the next tool observation. A fixed decision policy π(a∣h)π(a h) gives action probabilities. A tool-response rule qu(r∣h,a)q_u(r h,a), with controlled external condition u, gives observation probabilities. Then at∼π(⋅∣ht),rt+1∼qu(⋅∣ht,at),ht+1=append(ht,at,rt+1).a_t π(· h_t), r_t+1 q_u(· h_t,a_t), h_t+1=append(h_t,a_t,r_t+1). (1) The append operation adds the action and delivered observation to the history. A return-withholding intervention changes the observation delivered to the next call while preserving the request. Unrolling these decisions in time gives forward event dependencies even though the workflow has feedback. Fix the policy, environment rule, observation intervention and window, and compare the resulting readout laws for histories with the same z. The sampled actions need not coincide. This supplies the same representation question as a fixed-input comparison: does the chosen description preserve the specified response distribution? The differentiable formulas below and finite categorical comparisons are distinct specializations of that question. An initial summary sufficient for one window need not predict its own next state; autonomous reduced dynamics require additional closure conditions. 2.2 Forward analysis: propagation and readout response A source history produces a trajectory and its measured observables over the chosen window. Finite contrasts compare the retained states and readouts under two source histories, with preparation and other inputs fixed. Let utu_t collect the controlled source coordinates and let OT(sT)=(O1,T,…,Od,T)∈ℝdO_T(s_T)=(O_1,T,…,O_d,T) ^d collect d≥1d≥ 1 numerical observables at time T. Window summaries such as amplitude and phase require the accumulated measurements in sTs_T (SI Section S1.2). Stochastic outputs are compared through a specified conditional statistic or output distribution. In response formulas, Δ denotes finite contrasts and δ tangent perturbations. A first-order response is the differentiable specialization. The derivative DxFD_xF maps a small change in x to the first-order change in F: it is a Jacobian in finite-dimensional coordinates, or a Fréchet derivative on the specified function space. On a declared time grid write the transition as st=Ft(st−1,ut)s_t=F_t(s_t-1,u_t), with At=DsFtA_t=D_sF_t and Bt=DuFtB_t=D_uF_t evaluated along the baseline trajectory. Differentiability of the transition, coarse-graining and readout maps on the perturbation space gives δst δ s_t =Atδst−1+Btδut, =A_tδ s_t-1+B_tδ u_t, (2) δOT δ O_T =DsOT(sT)∑t=1TΦ(T,t)Btδut, =D_sO_T(s_T) _t=1^T (T,t)B_tδ u_t, Φ(T,t) (T,t) =AT⋯At+1. =A_T·s A_t+1. (3) The retained-state response applies DsGfD_sG_f to the same propagated displacement. Each term follows one perturbation: BtB_t injects it, successive transitions AtA_t propagate it, and DsOTD_sO_T converts the terminal displacement into a measured change. Here Φ(T,T)=I (T,T)=I, the identity operator. The formula fixes the initial state; varying its history adds (DsOT)Φ(T,0)δs0(D_sO_T) (T,0)δ s_0. If the endpoint is a differentiable function O¯T O_T of the retained state, then OT(sT)=O¯T(Gf(sT))O_T(s_T)= O_T(G_f(s_T)). The bar distinguishes this function of zTz_T from the function of sTs_T; it does not denote an average. The chain rule gives DsOT(sT)=DzO¯T(zT)DsGf(sT).D_sO_T(s_T)=D_z O_T(z_T)\,D_sG_f(s_T). The rightmost derivative maps full-state to retained-state perturbations; the other maps them to the readout. This composition is the hypothesis tested against the independently measured response. For the identity representation Gf(s)=sG_f(s)=s, its derivative is I; a graph projection must separately retain the histories and records required by the readout. Differentiability is additional to Lipschitz well-posedness; nonsmooth points require finite contrasts or specified one-sided responses. Step refinement compares slopes at two perturbation sizes to assess a first-order interpretation. Figure 2 shows why state measurements and selective state interventions answer different questions. Figure 2: State measurement and causal intervention answer different questions. Here u is the source, z the retained state and O the endpoint readout. a, In z=gpu,O=grz=g_ pu,\ O=g_ rz, gpg_ p is the propagation gain and grg_ r the readout gain. Halving either from gp=gr=1g_ p=g_ r=1 gives the same endpoint O=u/2O=u/2, but different states. Restoring the altered gain restores O=uO=u; ordinary state-space identification gives the same decision. b, The models O=2zO=2z and O=2uO=2u, both with z=uz=u, have identical source-response records. Independently setting z=vz=v while holding u fixed (a clamp) separates them. These are analytic models, not AI measurements (SI Section S1.3). The time at which a state is tested matters: it can be the end of the window or its beginning. For a terminal state zT=Gf(sT)z_T=G_f(s_T), the hypothesis is that OT=O¯T∘GfO_T= O_T G_f. For a preparation history h, let z0=ψ(h)z_0=ψ(h) be the candidate summary before a continuation v: a fixed input sequence, or fixed external response and observation rules for the fixed adaptive policy in Section 2.1. Write RT,v(h)R_T,v(h) for the readout obtained by evolving that history through the window. A readout function HT,vH_T,v depending only on the summary would satisfy RT,v(h)=HT,v(ψ(h)).R_T,v(h)=H_T,v(ψ(h)). It asks whether the preparation summary preserves the distinctions needed for that future readout. It does not assert that the summary evolves autonomously. For a stochastic system, replace readout equality by equality of the specified statistic or conditional output distribution. Comparisons always hold v and the system fixed. In the running example, the pending-request snapshot is tested before the returns arrive, rather than against an output already stored at the endpoint. 2.3 Inverse analysis: is the retained information sufficient? Exact sufficiency requires histories with the same candidate state to give equal specified readouts or output laws under common continuations. A resolved difference rejects sufficiency. Finite comparisons can instead bound agreement at a specified resolution or leave adequacy unresolved. For differentiable responses, a first-order test asks whether a source direction invisible to the state remains visible to the readout. On a finite-dimensional measured source space TU_T, let z,TK_z,T and O,TK_O,T be the first-order source-to-state and source-to-endpoint maps. For a linear map K, kerK K contains inputs mapped to zero, and ImKImK contains its possible outputs. A linear readout map ℋT(1)H_T^(1), with superscript (1)(1) denoting first order, exists exactly when [∃ℋT(1):O,T=ℋT(1)z,T]⟺kerz,T⊆kerO,T. [∃\,H_T^(1):\,K_O,T=H_T^(1)K_z,T ] _z,T _O,T. (4) The map is unique on Imz,TImK_z,T, the reached state displacements. To challenge the representation with this test, the design must include a source direction that leaves the state unchanged to first order. Such a direction lies in kerz,T _z,T and requires dimT>rankz,T _T>rankK_z,T. At full column rank there is no such direction, and the factorization holds automatically. First-order agreement alone does not establish agreement for finite changes. For continuously differentiable (C1C^1) maps between finite-dimensional coordinate spaces, constant rank of the source-to-state map and kernel inclusion throughout a neighborhood give local nonlinear factorization. Without constant rank, z(u)=u2z(u)=u^2, O(u)=u3O(u)=u^3 obey the pointwise derivative criterion but merge opposite inputs with different endpoints. Exact neighborhood sufficiency instead requires equal retained states to imply equal endpoints; stochastic readouts require repeated controlled continuations and an appropriate distributional comparison. The proof and analytic diagnostic controls appear in SI Section S1.3; the delayed-history counterexample is given in Section 3. Apply the criterion after checking response resolution and, for a first-order claim, step refinement. The design must supply a resolved direction in the state-response kernel or a controlled same-state comparison. For approximately matched states and a deterministic readout, let εz _z bound the true state difference and εO _O the error in the measured readout contrast ΔO O. A candidate readout O=h(z)O=h(z), with independently justified Lipschitz bound LhL_h (at most LhL_h units of output change per unit of state change), requires ‖ΔO^‖≤Lhεz+εO.\| O\|≤ L_h _z+ _O. (5) The right-hand side is the output change allowed by imperfect state matching and measurement error. A violation rejects this bounded readout class; with exact matching, it rejects the representation for the compared histories. For an approximate match, check the sensitivity assumption as well as the state. Local support requires adequate comparison coverage, contrast intervals within fixed equivalence bounds and a detectable positive control: a deliberately varied input whose readout change is resolved beyond measurement uncertainty. Stochastic readouts require corresponding statistical or distributional bounds. Unresolved comparisons remain undecided; different directions and readouts can receive different judgments. Equivalence margins and positive-control thresholds are study-specific design choices, fixed before the observations used for those decisions. The response framework supplies no universal numerical values for them. Sample size, output alphabet and simultaneous error allocation determine the attainable precision of a chosen confidence bound. Even identical empirical distributions can remain unresolved when that precision is insufficient for the chosen agreement margin. For categorical outputs, total variation (TV) is the largest difference in probability assigned to an output event. A matched-state pair has bounded agreement when its TV upper bound is at most the study’s tolerance; a lower bound above that tolerance rejects it; overlapping bounds leave it unresolved. Informative support also requires the study’s coverage and positive controls. Methods selected after observing responses provide exploratory evidence and leave the original decision classifications unchanged. Table 2 translates these conditions into experimental choices. Repeated measurements reduce uncertainty, but cannot supply a direction in the state-response kernel or a matched-history pair absent from the design. Table 2: Response comparisons guide experimental design. Judgments concern a specified readout, perturbation class and window; state-matching error and measurement uncertainty must be bounded. First-order claims additionally require slopes consistent across perturbation sizes. Judgments may coexist across directions. Stochastic readouts require bounds on differences in the chosen statistic or conditional output distribution. Observation Supported conclusion Next experiment Adequate direction coverage; readout-difference bounds, including errors, within the fixed equivalence margin; detectable positive control. Bounded support on the tested comparisons at the declared resolution. Extend directions, readouts or windows to test the scope of support. Exactly equal retained states give a readout difference beyond measurement error. Reject sufficiency for these histories under the declared conditions. Separate the violating histories in a revised representation; test it with system and interface unchanged. Approximately matched states violate the independently justified bound combining readout sensitivity and measurement error. Reject the bounded readout class, not the representation alone. Check the readout class and independent sensitivity bound; tighten state matching. Neither an informative direction in the state-response kernel nor a matched-state comparison is resolved. The design leaves adequacy unresolved. Change source directions or compare different histories at matched states. Response is detected; slopes fail the step-refinement tolerance. Finite contrasts; first-order interpretation is unsupported at these steps. Test finer steps or another readout; recheck resolution. A task-required contrast is absent despite preserved input records. Test how the program or model uses the supplied records. Compare changes to the program or model with controls; remeasure state and readout. 3 Forward response in physical systems Known physical dynamics make the forward question directly testable: does the measured response agree with the response calculated from the equations of motion? A field with local memory and delayed sources supplies the dynamical setting; stationary susceptibility follows when additional closure and stability conditions hold. Simulated delayed consensus calibrates amplitude and phase against an analytic prediction, while a polar flock compares nonlinear steady-state readouts. These tests use standard delay-system and active-matter models 18, 39, 55, 53, 31. The shared response description fixes the roles of source, retained state and readout; each system supplies its own equations and noise law. 3.1 History-dependent state and dynamics Neural-population and neural-field models describe collective activity through local interactions and spatial coupling 59, 2, 6. To include local memory and delayed sources, consider the controlled semilinear field ∂tct=Lct+∑i=1NJi(ct,mi(t))+∑ℓ=1NτEℓyℓ(t−τℓ). _tc_t=Lc_t+ _i=1^NJ_i(c_t,m_i(t))+ _ =1^N_τE_ y_ (t- _ ). (6) Here ct∈L2(X,ℝq)c_t∈ L^2(X,R^q) has q channels on a spatial or graph domain X. The operator L describes how the field spreads or relaxes between source contributions; a diffusion–decay operator is one example. The N local sources JiJ_i depend on the field and local memories mim_i. Each of the NτN_τ external channels injects a prescribed signal yℓy_ through EℓE_ after delay τℓ _ . The local source can itself be resolved into physical operations. Let RiR_i read the field and memory, πi _i compute a local update, WiW_i convert that update to a source, and GiG_i place it into the field’s sites and channels. Then Ji(c,mi)=GiWi(πi(Ri(c,mi))).J_i(c,m_i)=G_iW_i\! ( _i(R_i(c,m_i)) ). (7) This separates injection from propagation. A physical component senses and drives the medium; a workflow component reads a record and writes an update. Each realization specifies these maps. The coupled response includes changes in field-dependent sources and memories as well as propagation generated by L. Delay determines what must be included in the state. In consensus with coupling delay τ, retain the history segment ξt(θ)=c(t+θ)ξ^t(θ)=c(t+θ), −τ≤θ≤0-τ≤θ≤ 0, together with any local memories. Two preparations can agree at c(t)c(t) while their past fields produce different forcing later in the observation window. This internal feedback belongs to the modeled dynamics, with its history suppressed in the displayed arguments of JiJ_i. The external-input sum instead describes signals whose generators lie outside the modeled system. In a delayed-tool workflow, earlier requests similarly supply context for later returns; whether a particular summary preserves that context is a representation question. The inverse criterion exposes what a present-value description can lose. For the delay equation x˙(t)=−ax(t)−kx(t−τ)+u(t),k≠0,τ>0, x(t)=-ax(t)-kx(t-τ)+u(t), k≠ 0, τ>0, let the candidate state retain only x(t)x(t). Here a is the local relaxation coefficient, k the delayed coupling, and u the applied input. Two allowed history preparations with the same current value but different x(t−τ)x(t-τ), under the same subsequent input, have different current derivatives. In tangent form, δx(t)=0δ x(t)=0 and δx(t−τ)≠0δ x(t-τ)≠ 0 give δx˙(t)=−kδx(t−τ)≠0δ x(t)=-kδ x(t-τ)≠ 0. The readout O=x(t+Δ)O=x(t+ ) therefore separates the preparations for sufficiently small Δ>0 >0. The present value omits a response-relevant history distinction. In the delayed-tool example of Section 5, the corresponding comparison holds a snapshot fixed while changing which earlier request a later return answers. Both examples test history compression using the inverse criterion of Section 2. SI Section S2.1 specifies the source and memory maps; SI Sections S2.2 and S2.3 give the history dynamics and existence conditions. Finite-window measurements follow this augmented state directly. Autonomous reduction and steady susceptibilities add the closure and stability assumptions stated next. 3.2 From finite-window response to stationary susceptibility The finite-window response also connects to the familiar displacement of a stable operating state under a persistent source. Suppose the retained coordinates admit a closed, finite-dimensional equation z˙=f(z,u) z=f(z,u), with f continuously differentiable, and let f(z∗,u∗)=0f(z_ ,u_ )=0. Controlled closure means that these coordinates determine their evolution for the allowed inputs on a common invariant domain, with compatible initialization. Invariance at each fixed input alone does not supply this property; explaining one endpoint readout is also a separate condition. Define the drift Jacobian Az=Dzf(z∗,u∗)A_z=D_zf(z_ ,u_ ) and source Jacobian Bz=Duf(z∗,u∗)B_z=D_uf(z_ ,u_ ). If all eigenvalues of AzA_z have negative real part, a constant tangent source δuδ u, starting from zero initial displacement, gives δz(t)=∫0teAz(t−s)Bzδus⟶χ∗δu,χ∗=−Az−1Bz.δ z(t)= _0^te^A_z(t-s)B_zδ u\,ds\ \ _ δ u, _ =-A_z^-1B_z. (8) Each contribution is injected by BzB_z and propagated by the linearized dynamics; stability makes their accumulated effect converge. For a differentiably factored readout O=O¯(z)O= O(z), its stationary response is δO=(DzO¯)χ∗δuδ O=(D_z O) _ δ u. A time-independent discrete transition with Jacobians A,BA,B and spectral radius ρ(A)<1ρ(A)<1 gives the corresponding sum χ∗G=∑k≥0AkB=(I−A)−1B _ ^G= _k≥ 0A^kB=(I-A)^-1B. With exact discretization under an input held constant during each time step, the two susceptibilities agree (SI Section S2.2). Thus finite-window propagation and stationary susceptibility belong to the same response description under their respective assumptions. For an AI realization, the stationary interpretation requires the same closure and stability conditions; finite contrasts remain applicable without that interpretation. 3.3 Operating boundaries depend on the readout An operating criterion specifies acceptable values of the scalar or vector readout OTO_T defined in Section 2: a threshold for one observable, or a region for several, such as amplitude, phase and spatial variance. The requirement defines the boundary; dynamics determine whether loading carries the readout across it. Readout dimension and units follow the measurements and need not match retained-state dimension. Neither two components nor nesting is required. To distinguish continued operation from a stricter endpoint criterion, fix an experimental family f: its instances, interface, stopping rule, loading coordinate and readouts. The loading coordinate parametrizes the conditions under which the readouts are compared. When the two readouts factor through a candidate state z∈Zfz∈ Z_f, write Pf,Qf:Zf→[0,1],0≤Qf(z)≤Pf(z)≤1.P_f,Q_f:Z_f→[0,1], 0≤ Q_f(z)≤ P_f(z)≤ 1. Here PfP_f represents the outer criterion for continued operation and QfQ_f a stricter condition contained within it. Nesting concerns the meanings of the conditions; numerical ordering alone does not establish it. Their reduced coordinates are wf:=Pf,rf:=Qf/Pfwhere Pf>0.w_f:=P_f, r_f:=Q_f/P_f P_f>0. (9) Thus wfw_f measures remaining workability and rfr_f the inner readout relative to the outer one. The ratio is a conditional correctness fraction only for nested measures on the same population. Criteria that do not factor through the candidate state remain readouts of the full measured system. Write w=wfw=w_f, r=rfr=r_f for one family. The four regimes are exact conditional on workability (w>0,r=1w>0,\ r=1), partially exact (w>0, 0<r<1w>0,\ 0<r<1), workable but inexact (w>0,r=0w>0,\ r=0), and failed (w=0w=0, with r undefined). At r=1r=1 the two readouts are equal; w separately records workability. Measured boundary assignments require stated uncertainty and tolerances. Readout level sets define boundaries; dynamics determine how a loading path approaches them. Encountering all four regimes in order requires additional regularity, monotonicity and separated-crossing assumptions. Here monotone loading excludes re-entry into a departed regime; the path and loading coordinate are specified before inspecting the crossing order. Coincident thresholds can skip regimes, and nonmonotone recovery remains compatible with nested readouts (SI Section S2.4.1). Source perturbations can change either readout; their responses are measured separately. The readout pair determines neither state dimension nor sufficiency, and response magnitude alone does not measure intervention cost. Figure 3 separates the allowed regimes from the additional assumptions needed for an ordered sequence of boundary crossings. Figure 3: Readout-regime map and boundary order under monotone loading. a, The coordinates are workability w=Pfw=P_f and the readout ratio r=Qf/Pfr=Q_f/P_f for Pf>0P_f>0. The four local readout regimes comprise one open partially exact region, the line r=1r=1 (exact given workability), the line r=0r=0 (workable but inexact), and the failed case w=0w=0. Hollow endpoints exclude w=0w=0 from the ratio chart: failure is specified without a value of r. The plotted path is schematic. SI Section S7.6 discusses retrospective coordinate examples, which are not encoded by this path. b, Along a specified monotone loading path whose boundary encounters cross between adjacent readout regimes, the four-step sequence holds when all three encounters occur with sEW<sWR<sROs_EW<s_WR<s_RO. Here sEWs_EW is the first encounter with the exact/partially exact boundary, sWRs_WR with the partially exact/workable-but-inexact boundary, and sROs_RO with the workable/failed boundary; an unencountered boundary has parameter +∞+∞. If sRO<∞s_RO<∞ and sRO≤sWRs_RO≤ s_WR, the path can pass directly from the partially exact region to the failed boundary. Nested readouts fix the allowed boundary cases but do not determine their encounter order. Nested readouts can be same-population fractions or single-trial indicators. A binary indicator is not differentiated: response measurements use finite contrasts or differentiable observables, such as amplitude and phase, with thresholds calibrated to the binary criteria. SI Section S2.4.1 gives the boundary definitions and conditional crossing results. In the consensus experiment below, PfP_f indicates continued admissibility and QfQ_f its stricter endpoint criterion. In the flock, the same family-local labels denote polarization and angular alignment: these are distinct, nonnested readouts. The distinction allows the same response framework to compare both, without assigning the nested regime labels to the flock. 3.4 Calibrating predictions in delayed consensus A sinusoidal drive supplies an analytic reference for measured amplitude and phase. The experiment checks this response over a narrow range of coupling delays near one operating point. This weak-delay calibration tests the driven response. Establishing history necessity from measurements would additionally require preparations with equal current values and different histories. Let ci(t)c_i(t) be the scalar activity at site i of a periodic, degree-four ring with N=64N=64 sites. Each site relaxes at rate a, receives its neighbors’ earlier activity with coupling strength K and delay τ, and is driven by a uniform sinusoidal source of amplitude F and angular frequency Ω , with site-noise amplitude σ: c˙i(t)=−aci(t)+K|Ni|∑j∈Nicj(t−τ)+Fcos(Ωt)+σξi(t). c_i(t)=-a\,c_i(t)+ K|N_i| _j∈ N_ic_j(t-τ)+F ( t)+σ _i(t). (10) Here NiN_i is the set of four neighbors and ξi _i are independent unit-intensity white noises. The delay calibration uses a=1.0a=1.0, K=0.40K=0.40, F=0.25F=0.25, Ω=0.35 =0.35 and σ=0.015σ=0.015, with Ωτ=0.1505 τ=0.1505–0.16100.1610. The ring averaging matrix is symmetric with unit row and column sums. Consequently the spatial mean m=N−1∑icim=N^-1 _ic_i closes exactly, including the projected noise: m˙(t)=−am(t)+Km(t−τ)+Fcos(Ωt)+ηm(t),ηm(t)≡σN∑i=1Nξi(t). m(t)=-a\,m(t)+K\,m(t-τ)+F ( t)+ _m(t), _m(t)≡ σN _i=1^N _i(t). (11) This closure removes the transverse spatial modes from the mean equation, but retains its delay history. For the stated parameters, a>|K|a>|K| ensures delay-independent stability of the normalized network modes. After transients, the deterministic component or ensemble mean has the form mdet(t)=ReFχ(Ω,τ)eiΩtm_ det(t)=Re\Fχ( ,τ)e^i t\, where harmonic balance gives χ(Ω,τ)=(a−Ke−iΩτ+iΩ)−1.χ( ,τ)= (a-Ke^-i τ+i )^-1. (12) The factor e−iΩτe^-i τ is the phase shift acquired by the delayed coupling. It changes both the amplitude and phase of the driven response. This frequency-dependent susceptibility complements the constant-source, stable-branch limit in Eq. (8). To compare this prediction with a finite-window measurement, a lock-in estimator extracts the part of the mean oscillating at the drive frequency over a window of duration TwinT_ win. The readout combines its amplitude and phase lag with the transverse spatial variance: Z^=2Twin∫windowm(t)e−iΩtdt,cons=(,ϕ,V),=|Z^|,ϕ=|argZ^|,V=1NTwin∫window∑i(ci(t)−m(t))2dt. gathered Z= 2T_ win _ windowm(t)e^-i t\,dt, y_ cons=(A,φ,V),\\ A=| Z|, φ=| Z|, V= 1NT_ win _ window _i (c_i(t)-m(t) )^2\,dt. gathered (13) Amplitude and phase measure the coherent response; V records spatial fluctuations that the mean discards. These three observables summarize a trajectory, rather than forming an autonomous three-dimensional state. The harmonic targets are th(τ) _ th(τ) =F[(a−KcosΩτ)2+(Ω+KsinΩτ)2]−1/2, =F [(a-K τ)^2+( +K τ)^2 ]^-1/2, (14) ϕth(τ) _ th(τ) =atan2(Ω+KsinΩτ,a−KcosΩτ). =atan2 ( +K τ,\;a-K τ ). For a local calibration over the narrow interval τ∈[0.43,0.46]τ∈[0.43,0.46], 100 noise realizations at each of 13 delays supply the mean amplitude and phase compared with these targets (Table 3). Most of the phase discrepancy comes from the conjugate-frequency contribution within the finite rectangular window. SI Section S3.1 separates this bias from the numerical correction and gives the update, initial history, seeds and provenance limits. Spatial concentration compares squared peak amplitude with transverse variance; it does not determine state dimension. Table 3: Consensus response errors and criterion crossings. Response errors compare 100-seed means with harmonic targets (maximum over 13 delays in [0.43,0.46][0.43,0.46]), including finite-window estimator bias; relative amplitude error uses the harmonic amplitude as denominator. A is peak amplitude and V transverse variance. Drive F and noise σ use model units. Subscripts P,QP,Q denote admissibility and its stricter amplitude/phase criterion, respectively. Crossing values interpolate where the fraction of the same 50 seeds satisfying each criterion crosses one half; they are sample statistics, not identified population boundaries or critical points. Comparison Recorded result Coherent response Relative amplitude error ≤2.51×10−3≤ 2.51× 10^-3; absolute phase error ≤0.74∘≤ 0.74 . Spatial concentration 2/(2+V)≥0.999A^2/(A^2+V)≥ 0.999; transverse variance is small relative to squared peak amplitude, without determining state dimension. Decreasing drive FQ≈0.163F_Q≈ 0.163, FP≈0.118F_P≈ 0.118. Increasing noise Sample crossings σQ≈0.138 _Q≈ 0.138, σP≈0.310 _P≈ 0.310; the early QfQ_f crossing is consistent with finite-sample phase fluctuations. Predicting a response and judging acceptable operation are separate questions. Retrospective sweeps compare continued admissibility with a stricter amplitude/phase requirement as drive or noise changes. At τ=0.43τ=0.43, both sweeps hold the original criteria fixed: Pf P_f =[stable,≥0.16,V≤0.05], =1\! [stable,\ A≥ 0.16,\ V≤ 0.05 ], (15) Qf Q_f =Pf[≥0.22,ϕ≤35∘]. =P_f\,1\! [A≥ 0.22,\ φ≤ 35 ]. Here “stable” denotes finite-valued numerical integration through the horizon with a nonempty observation window; it is distinct from the analytic stability condition a>|K|a>|K|. The stricter criterion requires both a larger coherent response and a small phase lag. The sweep denominator comprises 26 control values with 50 matched seeds each, separately from the delay calibration. All 1,300 recorded sweep trajectories satisfy the numerical condition, while 442/1,300 remain admissible but fail the stricter endpoint criterion (Qf=0<PfQ_f=0<P_f). This counts trajectories between the two operating criteria. The noise sweep illustrates why a sample crossing must be distinguished from a population boundary. All 179 trajectories with Pf=1,Qf=0P_f=1,Q_f=0 fail the phase criterion. Yet at σ=0.15σ=0.15, where only 24/5024/50 seeds pass that criterion, the linear Gaussian dynamics and numerical estimator give a population phase-pass probability of approximately 0.5830.583. It remains above one half throughout the recorded noise range. Reusing the same noise realizations strongly correlates neighboring measurements, so their early crossing σQ≈0.138 _Q≈ 0.138 is consistent with finite-sample phase fluctuations. The complete QfQ_f criterion can still cross one half when its variance constraint becomes active. SI Section S3.1 gives the complete counts, pointwise intervals and conditional bootstrap analysis; these quantify sampling variation without identifying a population phase boundary. Figure 4: Physical response calibration and complementary steady-state readouts. a, Time-centered field on a 64-site delayed ring over four drive periods; color is in model units. b, Fractions of its squared norm in spatial Fourier modes, with 99.81%99.81\% in the uniform mode k=0k=0. Panels a,b use one seed at τ=0.46τ=0.46; these weights differ from the peak-amplitude and quadrature weights in SI Section S3.1. c,d, The separate 13-delay calibration compares 100-seed means (circles), harmonic predictions (solid lines) and the finite-window estimator applied to the harmonic signal (dashed lines). Relative to the harmonic target, amplitude and phase have maximum relative and absolute differences 2.51×10−32.51× 10^-3 and 0.74∘0.74 , respectively. The displayed differences include finite-window estimator bias, decomposed in SI Section S3.1; no uncertainty interval is inferred from the plotted means. e, Four-seed mean polarization PfP_f and field alignment within 30∘30 , QfQ_f, in the driven Vicsek flock. Guides mark one half and the first sampled noise levels below it, ηQ=0.35 _Q=0.35 and ηP=0.60 _P=0.60. The readouts are nonnested, despite numerical ordering on [0.15,0.70][0.15,0.70]; the sweep is not a boundary trajectory. Hollow squares expose coincident values without offsets. Shading spans the minimum and maximum of the four seed means; it is not a confidence interval and excludes within-run fluctuations. 3.5 Comparing readouts in a polar flock The consensus criteria are nested by definition. A driven polar flock permits comparison of distinct criteria without that restriction. Increasing angular noise tests how coherent motion and field alignment differ in a nonlinear steady-state response. Polarization measures agreement among particle headings; field alignment measures agreement with an external direction. In this realization, N=256N=256 particles move at constant speed v0=0.3v_0=0.3 in a periodic square of side L=8L=8. Their positions determine the neighbors NiN_i within radius rint=1r_ int=1, including the focal particle. Headings θi(t) _i(t) turn toward the local vector sum and a fixed field of strength h=0.6h=0.6 and direction θe^=0 _ e=0, subject to angular noise of amplitude η: θi(t+1)=Arg(∑j∈Nieiθj(t)+heiθe^)+ηξi(t),ξi∼(−π,π). _i(t+1)=Arg\! ( _j∈ N_ie^i _j(t)+h\,e^i _ e )+η\, _i(t), _i (-π,π). (16) Positions advance with the updated headings after each unit time step. The angular noises are independent across particles and steps. Their amplitude η is varied while the field stays fixed. Thus positions and headings supply the evolving state, local interactions propagate alignment, and angular noise competes with collective order. This model uses instantaneous neighbor headings; unlike consensus, its update has no explicit transport delay. Let δi(t)=Argei(θi(t)−θe^)∈(−π,π] _i(t)=Arge^i( _i(t)- _ e)∈(-π,π] be the wrapped angle from the field and let tol=30∘=π/6tol=30 =π/6 be the alignment tolerance. Instantaneous polarization and aligned population fraction are p(t)=|1N∑i=1Neiθi(t)|,q(t)=1N∑i=1N[|δi(t)|≤tol].p(t)= | 1N _i=1^Ne^i _i(t) |, q(t)= 1N _i=1^N1\! [| _i(t)| ]. (17) The reported readouts average each quantity over time and then four seeds: Pf=⟨p(t)⟩time⟩seed,Qf=⟨q(t)⟩time⟩seed.P_f= p(t) _ time _ seed, Q_f= q(t) _ time _ seed. (18) The modulus in p(t)p(t) precedes averaging. A coherent flock can turn away from the field while keeping polarization near one; QfQ_f instead asks how many particles point within the chosen tolerance. The readouts are therefore nonnested. Their numerical order on η∈[0.15,0.70]η∈[0.15,0.70] fails outside that interval, and Qf/PfQ_f/P_f is not a conditional fraction. The sampled half-values in Figure 4e quantify threshold-dependent separation, not a critical point, boundary trajectory or slow mode. Averaged values do not distinguish heading broadening from collective-direction fluctuations. SI Section S3.2 gives sampling, tolerance comparisons and the angular-distribution approximation. Together, the studies calibrate driven consensus near one operating point and compare flock readouts; history necessity follows from the analytical delay example. 4 Forward response in an AI event graph The physical studies predict responses from known dynamics. In an AI workflow, propagation through a prescribed graph can likewise be calculated, while the language model’s response to that graph must be measured. This forward test separates graph propagation from readout response and tests a first-order description at the measured perturbation sizes. Here a language model extracts events from visible records, and fixed rules map the checked events to a numerical graph retaining support and opposing evidence for candidate values. Separate model calls read a table derived from its state. Both model roles use Qwen2.5-14B-Instruct 42 with MLX 4-bit quantization. Measurement fixes the extracted graph, perturbs a source amplitude and compares the propagated state and model support for candidate values over a fixed window. The response is conditional on this graph; changing source text and repeating extraction is a different experiment. 4.1 From evidence records to graph dynamics To make the source encoding explicit, consider a constructed telescope task: choose an instrument, filter and observation block for a calibration plan. Instrument labels are alpha, beta and sigma; filters are broad, hydrogen-alpha and infrared; block labels are A, B and C. Three evidence statements specify beta’s stable lamp, the hydrogen-alpha filter and block C’s matching exposure. A fourth repeats infrared and block B while explicitly excluding them. These are authored constraints, not astronomical observations. The extractor receives the task, allowed values and evidence statements. Each event specifies its source record, update time td∈0,1,2,3t_d∈\0,1,2,3\, type, candidate targets and a confidence score. Checks enforce valid record references, times, types, targets and confidence values; they do not verify semantic accuracy. The event type determines the outgoing edges. A support event feeds a candidate’s support coordinate; contradiction and inhibition events feed its opposing-evidence coordinate; a coupling event divides its contribution equally between two candidates’ support coordinates. Contradiction and inhibition therefore share the same numerical action in this implementation. The graph builder, rather than the language model, sets edge weights from these rules. Confidence is recorded but does not set the imposed source amplitude. SI Section S1.4 defines this general map from records to numerical sources. The telescope extraction illustrates why format checks and faithful evidence interpretation are separate. It assigns support to beta, hydrogen-alpha and C at times 0, 1 and 2, but encodes the explicitly excluded infrared/B pair as a coupling at time 3. This recorded extraction remains fixed during the response experiment. The event assignments fix transport paths. The general transition FtF_t permits heterogeneous, time-dependent or nonlinear updates, with required memory retained (Section 2). This experiment uses a fixed graph with uniform retention and transport gains. For NeN_e extracted events, the numerical state zt∈ℝNe+18z_t ^N_e+18 contains one coordinate per event and two per candidate: support and opposing evidence for each of the nine allowed field–value pairs. The telescope instance has Ne=4N_e=4 and hence 22 coordinates. Let W be the fixed transport matrix: WijW_ij is the weight from source coordinate j to target coordinate i. An event’s outgoing weights sum to one; candidate coordinates have no outgoing edges. Let ede_d select event d’s coordinate, udu_d be its imposed amplitude, and btb_t the total deposit at update t. Let α∈(0,1)α∈(0,1) be the retention factor, β≥0β≥ 0 the transport gain, and I the identity. Starting from z0=0z_0=0, the updates are bt=∑d:td=tuded,zt+1=[(αI+βW)(zt+bt)].b_t= _d:t_d=tu_de_d, z_t+1=C\! [(α I+β W)(z_t+b_t) ]. (19) The dimensionless gains specify retention at each coordinate and transfer along edges; they are not complementary probabilities. The numerical range-control map C divides by the largest coordinate when it exceeds one, then clips coordinates to [0,1][0,1]. The reported experiment fixes α=0.72α=0.72, β=0.45β=0.45, and four updates t=0,1,2,3t=0,1,2,3. Neither range-control operation activates on its measured perturbation trajectories. Every baseline amplitude is ud=0.5u_d=0.5, independently of extractor confidence; these amplitudes are experimental inputs, not truth probabilities. Thus propagation along these trajectories is linear with A=αI+βWA=α I+β W. An amplitude perturbation enters through AedAe_d, because deposition precedes the update, separating source injection from propagation (Section 2). SI Section S7.1.1 gives its parameter-dependent state response. The measured model-readout results below concern the fixed coefficient pair; parameter robustness requires comparisons across coefficient settings. Equation (19) instantiates the finite-window response construction: the source amplitude enters a known transition, and its terminal effect is measured separately from the reader’s response. Its update index counts computational steps. The dynamics are prescribed and open-loop, without readout feedback or a claimed discretization of the continuous field equation. The representation tests in Section 5 use original request–return records instead of this graph apparatus. 4.2 Measuring propagation and model readout response To separate graph motion from model response, the reader receives the task, one candidate and a table extracted from z4z_4, containing each candidate’s support, opposing evidence and their difference. It receives neither the event-source coordinates nor the original evidence text. A fixed input transformation, chosen on separate development instances, divides candidate contrasts by 16 before the reader is queried. Let VjV_j be the three allowed values in field j. Each displayed quantity rjvr_jv, for value v∈Vjv∈ V_j, is mixed toward its within-field mean r¯j r_j: rjv(λ)=(1−λ)rjv+λr¯j,r¯j=13∑v∈Vjrjv,λ=0.9375,r_jv^(λ)=(1-λ)r_jv+λ r_j, r_j= 13 _v∈ V_jr_jv, λ=0.9375, (20) The measured response is conditional on this setting. A separate query asks whether each candidate is best supported. Write pjvYp^Y_jv for the model’s Y probability conditional on the two answer tokens Y and N, and let vj⋆v_j be field j’s reference value, used only in post-call scoring. At the terminal update T=4T=4, the scalar readout is OT=13∑j=13pjvj⋆Y∑v∈VjpjvY,T=4,O_T= 13 _j=1^3 p^Y_jv_j _v∈ V_jp^Y_jv, T=4, (21) Within each field, the denominator sums support across its three candidates; the ratio is the share assigned to the reference value. The readout averages these shares across fields; it is neither complete-plan accuracy nor a calibrated joint-choice probability. Extraction and graph construction are recorded before these reference values are used. Step refinement tests whether the model readout has a consistent local slope. Three development instances determine the reader-input transformation and response-detection thresholds; four different instances supply the response comparisons. For each measured graph, a fixed ordering selects three events without using endpoint labels. After extraction, perturb one selected amplitude at a time from 0.5 to 0.5±ϵ0.5±ε, first with ϵ=0.1ε=0.1, then with ϵ/2=0.05ε/2=0.05. Graph topology, event timing, other amplitudes and paired reader settings stay fixed. Record both the displacement of the complete graph state and the independently measured readout slope; subscripts d,+d,+ and d,−d,- indicate the increased and decreased amplitude of event d: Δzd=zd,+(4)−zd,−(4),g^d(ϵ)=Od,+−Od,−2ϵ. z_d=z_d,+(4)-z_d,-(4), g_d(ε)= O_d,+-O_d,-2ε. (22) Paired candidate queries use matched seed settings. Repeated-input controls check repeatability; routing controls check delivery of the numerical input. The latter restore the baseline graph state or disable edge transport; they check the numerical input path, not causal mediation. The telescope’s beta event illustrates the slope comparison. Amplitudes 0.4 and 0.6 give aggregate readouts Od,−≃0.8298O_d,- 0.8298 and Od,+≃0.9276O_d,+ 0.9276, as defined in Eq. (21). Their difference gives a slope of 0.489 per unit source amplitude; the halved step gives 0.639. Dividing the absolute slope difference by the larger slope magnitude gives 23.5% disagreement, within the fixed 25% tolerance. This is one direction’s readout response, not an improvement in task accuracy. The graph state responds in every tested direction, but the model readout satisfies the first-order test in only some directions. Across four tasks, all twelve Euclidean displacements of the complete terminal graph state exceed 0.05; nine readout slopes exceed 0.01 in magnitude, and two pass the step-refinement criterion. Four of the nine above-threshold directions reverse sign when the step is halved. In 355 of 432 perturbation calls (82.2%), the conditional probability of Y or N is at least 0.99. These descriptive saturation counts and finite-step comparisons cannot separate saturation from other nonlinear readout effects. Within each three-direction design, no nonzero linear combination of tested source contrasts preserves the complete graph state. The reader receives evidence only through the candidate table. These measurements characterize readout response after prescribed propagation, without a same-state comparison that could expose omitted response-relevant information. SI Section S7.1.1 gives sampling settings, the exact refinement criterion, denominators, controls and saturation diagnostics. 5 Inverse tests under fixed future inputs The forward studies predict or measure responses for a chosen state. The inverse question tests whether that state retains enough information: can histories with the same candidate state give different responses to the same future inputs? A request–return program first establishes exact sufficiency for a sensor–recipient association state. Language-model comparisons then test that state, its invariance to identifier and order changes, and a more detailed candidate. Each comparison keeps its program or model fixed while the analyst’s description changes. 5.1 Request associations distinguish identical snapshots The checkpoint test asks whether request associations can be omitted while preserving later readouts. Constructed histories and returns test a known message-processing program. Each execution combines one history and continuation; matched-state histories are compared under the same continuation. Selection and new executions test refinement within a supplied candidate set, rather than model performance on unknown tasks. At the observation cut t=0t=0, two requests have been issued to logical sensor channels S1,S2S_1,S_2. Each request associates its sensor with one recipient, A or B, and the association is bijective. A preparation history H=(σ,ν)H=(σ,ν) consists of the association orientation σ and the remaining request information ν. For σ=+1σ=+1, S1S_1 is associated with A and S2S_2 with B; for σ=−1σ=-1, these associations are reversed. The other records include request identifiers, issue times before the cut, and a text note unused by the program. The common checkpoint snapshot records a current value of zero and two pending requests. The future continuation C specifies sensor return values (u1,u2)(u_1,u_2), a query q∈A,Bq∈\A,B\, and an arrival schedule. Both returns arrive after the cut and by the fixed horizon T=3T=3, with returns at T processed before the final readout. The return value for each logical sensor is held fixed when comparing histories. Identifiers in return records match the original requests in each history; different identifier strings therefore do not change the logical continuation. The program joins returns to requests, assigns each value to its recipient, and reads the queried recipient’s quantity v. Let [⋅]1[·] equal one when its condition holds and zero otherwise. The vector observable is OT(H;C)=(v,[v≥θ]),θ=6.O_T(H;C)= (v,1[v≥θ] ), θ=6. (23) The threshold component is a function of the quantity, not an independent measurement channel. The example uses finite comparisons; the binary component does not require a derivative. Times specify the constructed event schedule in arbitrary units, not measured network latency. The program retains and reads the full original records in every comparison. Only the analyst’s candidate representation changes: zPz_P is the snapshot, zDz_D adds the two issue times, zJz_J adds the sensor–recipient association, and zFz_F retains the full preparation records. Extraction uses records available at the cut, before future values or the query are supplied. Sufficiency here requires a readout function hTh_T of the candidate state and continuation: OT(H,C)=hT(z(H),C)for every declared legal continuation C.O_T(H;C)=h_T (z(H),C ) every declared legal continuation C. (24) Keeping the future input explicit distinguishes a sufficient checkpoint description from autonomous dynamics. No candidate representation or prediction is supplied to the program, so refinement does not change its answers. For query A and sensor values (2,7)(2,7), the two orientations have the same zPz_P but outputs (2,0)(2,0) and (7,1)(7,1). This rejects zPz_P. Equal issue times in the two histories also reject zDz_D. The comparison identifies a distinction that must be retained; selecting its encoding remains a separate step. 5.2 Exact sufficiency of the association state The sensor–recipient association is sufficient because the program sends each sensor’s return to its associated recipient. Histories with the same association therefore give the same quantity and threshold indicator for every allowed continuation, regardless of request identifiers, pre-cut issue times, notes or arrival order. Both returns arrive by the observation horizon. Conversely, exchanging the associations changes the quantity reported for recipient A whenever u1≠u2u_1≠ u_2. Such values are allowed, so a sufficient representation must distinguish the two associations. The state zJz_J does exactly this: for the specified program, common snapshot and continuation class, it is the coarsest sufficient representation, with two states up to relabeling. This argument covers finite real returns under the specified schedule; the executions below check its implementation. The execution check asks whether a fixed selection rule finds the association state without retaining every history. Among the supplied candidates, it excludes same-state response violations, selects the fewest distinct states on the discovery histories and breaks ties by a fixed order. Selection uses those outputs; predictions use the analytical readout rule. Four discovery histories vary association orientation and unrelated notes. Snapshot descriptions merge histories with different responses; full history leaves no matched pair. The selection rule chooses the two-state association description zJz_J, then fixes it before new executions. SI Section S4.1 gives the inputs and checks. New executions test the selected state across changed notes, issue times and future inputs. Sixteen histories are evaluated under twenty-four continuations varying return values, arrival order and queried recipient, including equal returns that do not distinguish associations. These give 384 base evaluations. Each association state still merges eight distinct histories, so the test compares different histories rather than assigning each input its own state (Table 4). Table 4: Exact comparisons after representation selection. The constructed data comprise sixteen histories evaluated under twenty-four continuations, giving 384 base executions. State counts give the number of distinct candidate states. Each comparison is an unordered pair of histories with the same candidate state under one common continuation; a violation means their quantity/threshold readout vectors differ. These are dependent comparisons reusing executions, not independent tasks. Candidate representation States Comparisons Violations Snapshot zPz_P 1 2,880 1,024 Snapshot and issue times zDz_D 2 1,344 512 Snapshot and sensor–recipient association zJz_J 2 1,344 0 Full preparation history zFz_F 16 0 — The dash means no matched-state comparison is available, not successful validation. For zJz_J, 2(82)×24=1,3442 82× 24=1,344. Predictions agree with all 384 base executions. Fixing the association to S1→AS_1\!→ A instead makes 128 errors. Eight continuations have equal sensor returns (128 executions): their agreement cannot support snapshot sufficiency because the inputs do not distinguish the associations. The largest matched-snapshot quantity difference is 12. The controls test response to the queried sensor and invariance to the other. Starting from returns (3,11)(3,11), with S1S_1 arriving at time 11 and S2S_2 at time 22, each history and query receives a unit increment to either sensor. The quantity increases by one in all 32 queried-sensor controls and stays unchanged in all 32 unqueried-sensor controls. Predictions agree in all 64. Together with four discovery and 384 base evaluations, these are 452 deterministic program executions, not model calls or independent samples. The analytical argument establishes sufficiency for the specified program and continuations; new executions check its implementation after selection. The construction establishes an information requirement, not an algorithmic advantage over direct execution of the known program. Language-model adequacy and intervention value require separate measurements. 5.3 Testing the association state in a language model Associations suffice for the program in Section 5.1; whether they preserve a language model’s responses requires measurement. We replace the program with Qwen2.5-14B-Instruct (MLX 4-bit), which reads complete request–return records and reports the queried recipient’s quantity. Its checkpoint, instructions and sampling settings stay fixed. Candidate states and reference answers never enter its input. The restricted comparison tests reports for recipient A, chosen after development involving both recipients. It uses the checkpoint and candidates zP,zD,zJ,zFz_P,z_D,z_J,z_F of Section 5. Returns carry request identifiers, quantities and arrival times; requests precede the cut and both returns arrive by T=3T=3. These are constructed record times, not measured network delays. Model inference samples an integer quantity in a single-field JSON object; other completed responses enter INVALIDINVALID. All incorrect and malformed responses remain in the analysis. Joining returns to requests gives an independent task reference for scoring correctness. A preliminary screen finds no candidate that passes across both recipients. Four histories and two continuations receive four calls per input condition. The screen requires matched pairs with empirical total variation (TV) at most 0.150.15 on each, then minimizes state count with fixed ties. It supplies no equivalence confidence statement. Because no candidate passes, the planned 768 confirmation calls remain unrun. The failure is concentrated in queries about recipient B. All 16 responses querying A are correct; 5/16 querying B give the other recipient’s value, and an association-matched pair has empirical TV 0.750.75. Identifiers and notes vary together, while query, values and arrival order covary across continuations; the cause is not isolated. Shared issue times make zDz_D identical to zPz_P. SI Section S4.2 retains the inputs, all 32 development responses and 16 unscored sampling checks. A separate follow-up tests recipient A, a restriction chosen after development. Selection reuses its 16 responses to choose zJz_J; these are not independent confirmation. The representation and requests are fixed before new responses. Histories, values, arrival orders and controls retain the unexecuted broader design; only the second query changes from B to A, with corresponding reference quantities (SI Section S4.2). The follow-up tests whether opposite associations give different answers at the same snapshot, while matching associations give agreement despite other request changes. The prompt instructs matching returns to original request identifiers independently of arrival order. The follow-up compares four new histories combining both association orientations with two identifier/time/note preparations. Two continuation combinations query A, giving eight base conditions and four matched association pairs. They vary sensor values and arrival order together, not as independent factors. Four controls increment the queried or unqueried sensor for one preparation in each orientation. The design fixes 64 calls per condition (768 total), the margin and the stopping rule before responses, with no replacement histories or additional draws. SI Section S4.2 specifies the inputs. For matched histories, reference-answer frequencies bound the distance between full output distributions. Each condition fixes a history and continuation. Let Πi _i be the parsed-output law in condition i, including INVALIDINVALID, and aia_i its independently specified reference quantity. The count kik_i of responses equal to aia_i, among n=64n=64 draws, gives an exact one-sided binomial lower bound LiL_i on Πi(ai) _i(a_i)9. Allocating error 0.05/120.05/12 per condition gives simultaneous coverage at least 95%, conditional on independent draws from stationary output laws within each condition. Reused bounds require no independence between pair comparisons. Requested seeds and identical answers do not establish these sampling assumptions or their server implementation. On this simultaneous event, overlap at a common reference answer gives ai=aj⟹dTV(Πi,Πj)≤1−min(Li,Lj).a_i=a_j\; \;d_TV( _i, _j)≤ 1- (L_i,L_j). (25) If the reference quantities differ, Πj(ai)≤1−Πj(aj) _j(a_i)≤ 1- _j(a_j), hence ai≠aj⟹dTV(Πi,Πj)≥max0,Li+Lj−1.a_i≠ a_j\; \;d_TV( _i, _j)≥ \0,L_i+L_j-1\. (26) Responses unequal to the reference, including invalid outputs, remain in the denominator. Comparisons using these bounds can remain unresolved when two distributions concentrate on the same incorrect answer; task accuracy is therefore not equivalent to representation adequacy. At this fixed sample size, the bounds favor concentration at a common reference answer; identical dispersed empirical distributions may remain unresolved. Some non-reference responses are compatible with bounded agreement, so these bounds do not require complete empirical concentration. Bounded support requires every matched comparison to have an upper bound at most 0.150.15. A lower bound above 0.150.15 rejects the candidate on that comparison; otherwise the result is unresolved. For equal-reference pairs this rule based on reference-answer probabilities can support or leave unresolved, but cannot reject. The fixed snapshot comparison uses opposite sensor–recipient assignments with the first note and issue-time preparation, under returns (3,11)(3,11) with S2S_2 arriving first. That comparison and both queried-sensor increments additionally require lower bounds above 0.500.50. Both unqueried-sensor controls require upper bounds at most 0.150.15. Full history supplies no matched pair and is uninformative. The model retains access to information omitted by the snapshot, so the interface does not impose agreement at matched candidate states. The restricted comparison supports the association state: all 768 responses are valid and equal their references, and all queried- and unqueried-sensor controls pass. Opposite associations give quantities 3 and 11 under one continuation, and 13 and 1 under the other, while association-matched histories agree. Each condition has 64 reference responses, so Li=(0.05/12)1/64L_i=(0.05/12)^1/64. Equations (25) and (26) give conservative TV upper and lower bounds 0.08210.0821 and 0.83580.8358, meeting the agreement and detection thresholds. These are repeated measurements of four histories, not 768 independent tasks. Adequacy depends on which histories a candidate merges. Reusing these responses, the snapshot has eight rejected and four within-margin pairs; adding issue times leaves four rejected pairs. All four association-matched pairs lie within the margin (Figure 5a), whereas full history provides no matched pair and remains uninformative. This bounded support concerns four histories, two continuation combinations and recipient A. Identifier families covary with association orientation, and sensors appear in a fixed request order. The comparison therefore cannot distinguish the association partition from correlated identifier or position descriptions; this does not establish which information the model uses. Selection within a supplied library does not identify a unique hidden variable. The next comparison independently varies identifiers and request order to test extension beyond this restricted support. 5.4 Testing identifier and request-order invariance If associations suffice, changing request identifiers or list order while preserving those associations should leave the output distribution unchanged. A fixed comparison tests this requirement on eight new histories crossing two identifier schemes, two association orientations and two request orders. The identifier schemes are reused in both orientations, and the analyst records the association as a sensor-sorted map; changing request order therefore leaves this candidate state unchanged. Neither operation changes the model instruction or its access to the complete records. Both future-input combinations query A, with return references renamed consistently to preserve the sensor-level continuation. The fixed design gives 64 calls in each of 16 base conditions and four controls, forming 24 association-matched pairs. It allocates error 0.05/200.05/20 per condition and retains the earlier agreement, detection and control margins. SI Section S4.2 gives the input values, crossed factors and sampling procedure. The fixed bounds leave association-state agreement unresolved for some identifier and request-order changes: 21 comparisons lie within the equivalence margin and three remain unresolved. All 1,280 responses are valid. The unresolved comparisons share one input: with the first identifier scheme, the original request order and S2S_2 assigned to A, the first continuation gives the reference quantity 1717 in 21/64 calls and the other sensor’s quantity 55 in 43/64 calls. Changing only the identifier scheme or only the request order gives 1717 in 64/64 calls. Each of these empirical distribution differences is 43/6443/64. The specified reference-probability bounds give an upper limit of 0.82340.8234 and a lower limit of zero for these equal-reference comparisons; they therefore leave the comparisons unresolved, rather than formally rejecting the state. The crossed design fails the combined support criterion. Seven of the eight single-factor orientation comparisons resolve differences, rejecting both the snapshot and snapshot-with-times candidates. Queried and unqueried controls pass in one orientation and remain unresolved in the other, so support does not extend to independent identifier and order changes under this criterion. These new histories test extension beyond the restricted comparison, rather than retesting its original inputs. These behavioral observations identify neither internal computation nor invariance under arbitrary renaming and permutations. An exploratory analysis chosen after observing responses compares output categories without reference answers across all 20 conditions, using simultaneous component-probability bounds (SI Section S4.2). Each of the three unresolved association pairs has a conservatively rounded TV lower bound of 0.31210.3121, above 0.150.15. An association-state-only approximation assigns both histories the same output law; the triangle inequality makes its error on at least one input at least half their separation. This gives worst-case behavioral approximation error at least 0.15600.1560 on one input per pair, under this post-hoc family’s nominal 95% simultaneous coverage and the stated sampling assumptions (Section 8.3). The exploratory separation leaves the original classifications unchanged. Both analyses reuse the same observations, providing neither independent replication nor evidence of internal mechanism. Figure 5: Bounded agreement and its limits for one fixed language model. Candidate descriptions are computed by the analyst; the model always reads complete request and return records. The snapshot records the current value and pending-request count; the association state also retains the sensor–recipient mapping. a, Association-state comparisons in three distinct input sets. The restricted comparison supports four matched pairs. The crossed design has 21 within-margin pairs and three unresolved under its fixed reference-probability bounds; a separate post-hoc categorical analysis gives those three pairs a total-variation (TV) lower bound of 0.31210.3121. The comparison testing added identifiers and order supports all 112 pairs. Bars summarize individual bounds by classification; labels give comparison counts. b, That comparison rejects the snapshot but supports both the association state and its revision retaining identifiers and order. Displayed ranges span the individual simultaneous bounds, not confidence intervals for an average. The dashed line marks the fixed TV margin 0.150.15. Nominal 95% coverage is separate for each study and analysis, assuming independent, stationary replicates within each study; conditions within a replicate may be dependent. All studies query recipient A, a restriction chosen after development. Agreement in this comparison does not remove the exploratory evidence of crossed-study separation or demonstrate an advantage for the revision. 5.5 Testing an extended representation on new histories The crossed comparison motivates testing whether retaining identifiers and request order, in addition to associations, improves agreement with model responses. A new experiment tests that revision against the simpler association state, with all candidates, histories, comparisons and analysis rules fixed before new responses. The model, instruction and input schema remain unchanged; the recipient-A restriction is inherited from the broader development comparison. The fixed refinement criterion requires both agreement for the revised description and a violation of the simpler description on the new inputs. The combined criterion additionally requires the specified controls. Sixteen new histories cross identifier scheme, association orientation, request order and a joint timing/note change. The revised state retains the snapshot and ordered request-ID/sensor/recipient triples. It merges the two timing/note histories at each of eight states, giving 16 comparisons under two new continuations. Each revised-state pair has identical raw future returns; cross-identifier comparisons for the simpler state consistently rename return references. SI Section S4.2.1 specifies all inputs. The design specifies 256 calls in each of 32 base conditions and four controls. Simultaneous bounds compare the complete integer/invalid-output laws through marginal event probabilities and paired-output disagreements; reference answers only score correctness. The pairing requests one common seed across conditions in each replicate. Its validity assumes independent stationary replicate vectors with the stated marginal distributions, without requiring independence between conditions. SI Section S4.2.1 gives the error allocation, controls and the treatment of other integers. Both the association state and its revision pass on these new inputs; the fixed refinement criterion therefore does not establish a benefit from retaining identifiers and order. All 9,216 calls complete. The snapshot has 128 resolved violations among 240 comparisons; 64 occur with identical raw future returns. In contrast, all 112 association-state comparisons and all 16 revised-state comparisons have TV upper bounds at most 0.05310.0531, below the fixed 0.150.15 margin (Figure 5b). Both declared queried-sensor controls exceed the 0.500.50 detection threshold, and both unqueried-sensor controls lie within 0.150.15; these checks cover their specified layouts only. Task accuracy is assessed separately: 8,190/8,192 base responses and 1,024/1,024 control responses are correct, with no invalid outputs. Repeated calls are not additional tasks. These comparisons support omission of the specified timing/note differences. They do not resample the earlier input with 21/64 reference answers: its exploratory separation and the three originally unresolved comparisons remain unchanged. Model support concerns the tested histories; exact sufficiency concerns the specified request–return program. 6 Inverse tests under tool feedback The request–return tests fix later sensor inputs. We now test state adequacy when an action determines the next observation: the model selects a software test, receives its result and acts again. Release requires both unit and integration tests to pass for the current version; an earlier version’s pass leaves a current test outstanding. Fresh calls to Qwen2.5-14B-Instruct (MLX 4-bit), with simulated tools and releases, test this prescribed distinction over two decisions. The instructions give the complete rule and require current-version status assessment before action. This is a controlled test of specified information requirements, rather than discovery of an unknown requirement or a software benchmark (Figure 6). 6.1 Versioned records and action-dependent observations Two histories contain the same passing statuses but differ in which test belongs to the old version: one requires a unit test, the other an integration test. Before inference, the experiment program joins returns to requests by identifier and supplies completed records and pending requests. The model reports current-version statuses and chooses a test, wait, release or hold; the output format permits every status–action combination. The instructions also declare identifier spelling and record order irrelevant to version. Thus the measured invariances, like the version distinction, are prescribed. SI Section S5 gives the exact instructions. The chosen test returns an externally imposed pass or fail, hidden from the first call. Two fresh second calls share that first action: one receives the request and return, the other only the request. Policy, prompt and model stay fixed. The expected decisions are release after a pass, hold after a fail, and wait when the return is withheld. The readout is the action sequence, with invalid outputs retained; branches sharing the first action are dependent observations. The candidate states test whether version information is needed beyond passing statuses. Three analyst-only initial summaries retain the displayed statuses and pending count (status-only), the latest status per test regardless of version, or each current-version result including absence. Only the last distinguishes the missing tests; it omits identifiers and order. The model receives complete records throughout. After interface development, these task-derived candidates, the model, prompt, comparison rules and forecasts were fixed before confirmation. Eight histories cross the missing test with two identifier schemes and two request orders; version names stay fixed. Each imposed pass/fail condition has 64 episodes per history. SI Section S5 specifies construction and sampling. Figure 6: Prescribed version information tested through adaptive actions and tool feedback. The model is instructed to assess current-version statuses and then follow a complete decision rule; record joining is performed before the model call. a, The two focal passing results concern different software versions. The schematic omits a redundant predecessor result present in all confirmation inputs. The model selects the missing current-version test; its return changes the next decision. Withholding that return preserves the executed request. b, Intervals show conservative total variation (TV) bounds: the 32 violations for a version-omitting state (the second such state gives identical results), all 24 comparisons for the current-version state, and all 16 feedback controls. Each row shows their common bound; the fixed thresholds are 0.15 for state agreement and 0.50 for feedback controls. c, Dots show observed normalized Brier loss; lines bound expected loss, averaged over the 16 feedback conditions. Forecasts were fitted to development responses and fixed before new observations; the third predictor is the instructed decision rule. All predictions condition on the imposed pass/fail outcome and observation branch; they predict the model’s actions. Panels b and c share one nominal 95% confidence event, conditional on independent stationary repetitions within each condition–branch distribution. On that event, the expected loss difference between fitted forecasts is at least 0.11800.1180, conditional on the fits. The 1,024 episodes and 3,072 calls are repeated measurements of eight constructed histories. 6.2 Testing version information through action sequences Reproducing the observed action sequences requires distinguishing the missing current-version test; the two passing statuses alone are insufficient. Every call produced a valid action: request the missing test, then release after a pass, hold after a fail and wait when its return is withheld. To test whether each summary preserves action-sequence distributions, we use the total-variation rules in Section 2: agreement tolerance 0.15, control threshold 0.50 and 16 possible parsed sequences, including invalid outcomes. One nominal 95% confidence event covers all 32 distributions, crossing eight histories with pass/fail test outcomes and delivered/withheld returns. Repeats are assumed independent and stationary within each distribution; branches may be dependent. At this sample size, bounded agreement requires both samples to concentrate on the same sequence; this does not establish a deterministic law. For either version-omitting summary, the same 32 of 56 matched pairs are rejected: one set of missing-test contrasts, not independent evidence for two omissions. The current-version state has bounded agreement on all 24 matches, with maximum TV upper bound 0.14370.1437. The rejected pairs and all controls—eight pass-versus-fail and 16 return-withholding comparisons—have TV lower bound at least 0.71260.7126 (Figure 6b). These results support the predefined refinement and resolve a tool-return effect on the next action. The same separation implies worst-case behavioral approximation error ≥0.3563≥ 0.3563 for either rejected summary on at least one merged history, on the same confidence event (Eq. (32)). Forecasts quantify the predictive cost of omitting version information. Both are fitted to the development counts in SI Table S3, with identical estimation and smoothing, then fixed before confirmation. They condition on imposed pass/fail outcome and observation branch, without future sampled actions. Across 16 new feedback conditions, mean normalized Brier loss 7 (squared probability error scaled to zero–one) is 0.25590.2559 for status-only and 0.01810.0181 for current-version prediction (Figure 6c). The instructed rule has zero observed loss. These scores reuse the representation observations; they establish neither optimal compression cost nor an advantage over that rule. Readout choice changes the information requirement: both histories may end in release after different tests. An analysis prompted by interim observations retains only the final action, projecting the original confidence set. All 56 status-only matches then have TV upper bound 0.14370.1437, while the 32 full-sequence rejections remain unchanged. The supported summary concerns the initial history and omits pending-request information needed later; the comparison tests initial-state adequacy rather than autonomous dynamics. First actions were constant within each condition, so feedback was tested after that observed action. The scope is one model, a developed interface and two decisions. An earlier action-only interface returned RELEASE on all 32 initial calls. The successful interface jointly changes instructions, joins records before inference and requests status assessment; their separate effects were not tested. SI Section S5 reports the interface failures and other feedback pilots. 7 Inverse tests with scientific text The instructed tasks specify which history distinctions should matter. We now ask which content and ordering distinctions affect a fixed model’s judgments of scientific text. A workflow retrieves abstracts, selects excerpts and passes an ordered packet to a fixed reader. Comparing summaries of that packet tests whether document identity, excerpt content and document order account for its judgments, separately from scientific correctness. 7.1 Text inputs and candidate representations We vary excerpt content and document order separately to test what the representation must preserve. SciFact supplies claims, abstracts and annotated evidence 57; eight training-split claims follow an existing development list without response screening or substitutions (SI Section S7.4). For each claim, BM25 keyword ranking retrieves six abstracts. Deterministic selection takes the first three or last three sentences of each, producing 18-sentence packets in retrieved or reversed document order. Sentence order within each document stays fixed. The four conditions therefore vary excerpt content, document order or both. This uses real abstract text and new calls, but is a controlled comparison, not a held-out benchmark. Selection never uses annotations or answer labels. Annotations only track whether complete annotated rationales remain; shortened packets receive no correctness score because removal need not preserve the original label. The coarse state retains the claim, document identities and excerpt counts, merging all four conditions. The refined state adds exact excerpts and provenance but omits document order, giving two matched pairs per claim. Both are analyst descriptions; the reader always receives the ordered packet. Document identity cannot recover which sentences were passed, and matching unordered excerpts does not make the reader’s inputs identical. Qwen2.5-14B-Instruct (MLX 4-bit) receives the claim, titles and excerpts with document and sentence identifiers. Its task and instructions remain fixed. Outputs are support, refutation, insufficient evidence or INVALID for any response outside the one-letter format. Sixty-four fresh calls per condition estimate 32 four-category distributions: eight tasks and 2,048 repeated calls. The source is excerpt selection or ordering; the readout is the judgment distribution after that operation. These finite contrasts require neither a derivative nor a graph or feedback model. The total-variation comparisons of Section 2 use tolerance 0.150.15 and simultaneous 95% bounds across all four categories and 32 conditions, assuming fixed sample sizes and independent stationary repeats within each. Informative support additionally requires an INVALID-probability upper bound at most 0.150.15 and a same-claim excerpt-policy contrast with TV lower bound above 0.50.5. This positive control tests sensitivity to changed content, not to order. All comparisons are reported, including those without these controls. With 64 calls, the bound resolves agreement only for samples concentrated on the same category; dispersed samples require greater precision. SI Section S6 gives the intervals and complete observations. Figure 7: Excerpt content and document order make different demands on a state. Rows identify the eight SciFact claims by topic and original dataset identifier. a, Replace the first three sentences of each abstract with its last three, keeping document order fixed. The candidate description retains the claim, document identities and excerpt counts. b, Reverse document order while keeping all selected sentences fixed; the candidate description also retains the exact excerpts and their provenance, but omits document order. Leading and trailing excerpts are shown separately. Points are empirical total-variation distances; lines give simultaneous 95% bounds across the 32 conditions, each measured with 64 calls, assuming independent, stationary repeats within each condition. The dashed line marks the 0.150.15 agreement margin; an interval wholly to its left gives bounded agreement, wholly to its right rejects that margin, and an overlapping interval is unresolved. The dotted line in a marks the positive-control threshold: an excerpt-policy contrast must have a distance lower bound above 0.50.5. Only the leading-excerpt comparison for the vitamin claim (12) combines bounded agreement with a detectable positive control for the same claim. The distance measures changes in model judgments, not task accuracy. 7.2 Response differences from content and order The tested responses require distinctions in both excerpt content and document order (Figure 7). Neither candidate preserves all of these distinctions. The document-and-count state is rejected for three of eight claims, including claim 1003 whose lower bound of about 0.15780.1578 lies close to the 0.150.15 tolerance. Among 16 order comparisons at fixed excerpts, nine show bounded agreement, one rejects the tolerance, and six remain unresolved. Only one of the nine also passes the positive control; the other eight lack it. All 2,048 responses satisfy the format, which is separate from representation adequacy. The SciFact claim about folic acid, vitamin B12 and kidney-disease progression illustrates both requirements: changing excerpt content changes the judgments, and reordering the same excerpts changes them again. For the leading excerpts, all 64 replies in each order report insufficient evidence (TV upper bound ≤0.1250≤ 0.1250). Trailing excerpts in retrieved order instead produce 64 support judgments, resolving the content control (lower bound 0.7501>0.50.7501>0.5). Reversing those same trailing excerpts produces 12 refutation and 52 insufficient-evidence judgments, rejecting the unordered state (lower bound 0.7501>0.150.7501>0.15). Thus the leading-excerpt pair has bounded local support with a detectable control, while another pair for the same claim requires an ordering distinction. The immune-cell depletion claim (332) distinguishes agreement from correctness: all 256 replies report support although every packet retains complete source-annotated contradicting evidence. The matched pairs have bounded agreement but no detectable positive control, so do not meet the informative support criterion. The annotation discrepancy is not an independently adjudicated error rate for these shortened packets. A revision must distinguish the rejected content and order pairs. Neither tested description is adequate across the eight claims. A proposed revision requires new informative comparisons; assigning each input a unique state would remove the matched histories needed to test it. 8 Information required to preserve responses The response comparisons raise two general questions: what must a representation preserve across several systems, and what approximation error is unavoidable when it merges distinguishable histories? The results below constrain retained information for deterministic observables and stochastic output laws, without selecting a unique encoding. 8.1 Sufficiency for one or several systems A representation adequate for several systems must preserve every distinction required by any member. The systems may still produce different output laws. Let ℋ H be a nonempty finite set of preparation histories with the discrete sigma-algebra, and ℳ M a nonempty declared set of systems whose dynamics and observation rules remain fixed. Let C be a nonempty measurable set of continuation conditions admissible for every system–history combination. These specify subsequent input sequences, or external tool-response and observation rules for an adaptive policy included in the fixed system m (Section 2.1). Each system produces the chosen observable from a history and such a continuation; endogenous actions are integrated into its output law. Fix each system’s interface, readout, window and future noise law. For m∈ℳm∈ M, denote its conditional output distribution by m(⋅∣h,v) P_m(· h,v), where h∈ℋh∈ H and v∈v∈ C. For each system, assume these distributions are probability kernels measurable in (h,v)(h,v), on a common declared measurable output space. Deterministic outputs are included as point masses. Two histories are response-equivalent for this family when h∼ℳh′⟺m(⋅∣h,v)=m(⋅∣h′,v)for every m∈ℳ,v∈.h _ Mh P_m(· h,v)= P_m(· h ,v) every m∈ M,\ v∈ C. (27) A representation ψ:ℋ→Zψ: H→ Z, with its image equipped with the discrete sigma-algebra, is sufficient for every declared system exactly when ψ(h)=ψ(h′)ψ(h)=ψ(h ) implies h∼ℳh′h _ Mh . Necessity follows because a readout depending only on (ψ(h),v)(ψ(h),v) gives equal laws at equal states. For sufficiency, choose any history assigned to a state and use its output law: the implication makes the choice immaterial. Finite ℋ H, the discrete state image and the assumed kernel measurability in v ensure that these state-based laws are measurable. Equality of output laws partitions histories into response-equivalent classes. Every sufficient representation must distinguish these classes, which therefore give the coarsest sufficient partition, up to relabeling. Adding systems or allowed continuations adds equality requirements and can only refine it. The criterion makes the information preserved by a shared representation an explicit experimental target and specifies the comparisons needed to test it. Its mathematical basis is the established predictive equivalence principle 48, 26, applied here to declared response experiments. If all deterministic systems are allowed and two output values can be distinguished by a measurable event, adequacy requires an injective history encoding: a system can use any distinction that a compressed description discards. A useful compressed representation therefore has a specified readout and set of systems. 8.2 Finite evidence and transfer between systems Finite comparisons can reject a candidate or bound agreement on tested histories and continuations. An exact same-state violation identifies histories that a revised representation must separate; bounds within the declared tolerance establish agreement only on those comparisons. Informative local support additionally requires the comparison coverage and controls in Section 2. Tolerance-based agreement need not be transitive and does not define the exact equivalence classes above. A full-history description can give every preparation a different state, leaving no informative matched-history comparison. Agreement for one system can support agreement for another if their output laws are uniformly close. This requires an independently justified bound on the change of system, in addition to the measured within-state agreement. Using total variation distance dTVd_ TV as in Section 2, define the worst within-state discrepancy over the declared histories and continuations by dψ(m)=suph,h′∈ℋ,v∈ψ(h)=ψ(h′)dTV(m(⋅∣h,v),m(⋅∣h′,v)).d_ψ(m)= _ subarraych,h ∈ H,\ v∈ C\\ ψ(h)=ψ(h ) subarrayd_ TV\! ( P_m(· h,v), P_m(· h ,v) ). (28) If dψ(m0)≤εd_ψ(m_0)≤ and an independently justified uniform bound on the change in output law between systems satisfies suph,vdTV(m(⋅∣h,v),m0(⋅∣h,v))≤ρ, _h,vd_ TV\! ( P_m(· h,v), P_m_0(· h,v) )≤ρ, then the triangle inequality gives dψ(m)≤min1,ε+2ρ.d_ψ(m)≤ \1, +2ρ\. (29) Each history in a matched comparison contributes one term accounting for the change of system. The independently controlled discrepancy ρ is an additional assumption; within-state agreement in one model does not estimate it. The bound is sufficient, rather than necessary: systems with different output distributions can still preserve the same history distinctions. AI memory summaries, checkpoints and records passed between components thus have two separate experimental targets. A within-model test asks whether the summary preserves the required distinctions. A cross-model test fixes that summary and evaluates each declared model separately. Section 5.3 tests one model on a finite set of histories; the family criterion specifies the comparisons needed for several models. 8.3 A lower bound on error from merging histories If a state merges histories with separated output laws, every state-only approximation incurs error on at least one history. Fix a system m, continuation v and state value z. Let Fz=h∈ℋ:ψ(h)=zF_z=\h∈ H:ψ(h)=z\ be its nonempty set of matched histories, and abbreviate the output law as Ph=m(⋅∣h,v)P_h= P_m(· h,v). Any approximation using only (z,v)(z,v) must assign the same probability law R to every history in FzF_z. Define its optimal worst-case error and the diameter of that set of output laws by rψ(z,v) r_ψ(z,v) =infRsuph∈FzdTV(Ph,R), = _R _h∈ F_zd_ TV(P_h,R), (30) Dψ(z,v) D_ψ(z,v) =suph,h′∈FzdTV(Ph,Ph′). = _h,h ∈ F_zd_ TV(P_h,P_h ). (31) The infimum is over probability laws on the declared output space. For any common approximating law R, the triangle inequality gives dTV(Ph,Ph′)≤dTV(Ph,R)+dTV(Ph′,R)≤2supg∈FzdTV(Pg,R).d_ TV(P_h,P_h )≤ d_ TV(P_h,R)+d_ TV(P_h ,R)≤ 2 _g∈ F_zd_ TV(P_g,R). Taking the supremum over pairs and then the infimum over R yields rψ(z,v)≥12Dψ(z,v).r_ψ(z,v)≥ 12D_ψ(z,v). (32) Thus simultaneous lower bounds ℓhh′ _h for measured pairs in this state class and continuation give rψ(z,v)≥12maxh,h′ℓhh′r_ψ(z,v)≥ 12 _h,h _h , with the same coverage. This standard metric bound quantifies the cost of merging distinguishable histories. For two output laws, their equal mixture attains half the diameter; for larger sets the optimal error can be higher. Three point masses on distinct categories, for example, have diameter one and optimal worst-case error 2/32/3. The bound turns the experimental rejections into quantitative limits on any predictor using the rejected summary. For the SciFact order contrast (Section 7), TV separation ≥0.7501≥ 0.7501 implies approximation error ≥0.3750≥ 0.3750 for an unordered exact-excerpt state. For the request–return snapshot comparison (Section 5.3), separation ≥0.8358≥ 0.8358 gives error ≥0.4179≥ 0.4179. For the version-omitting initial states in the closed-loop workflow (Section 6), separation ≥0.7126≥ 0.7126 gives error ≥0.3563≥ 0.3563. Each downward-rounded bound applies to at least one merged history under its fixed system and continuation, inheriting that study’s simultaneous 95% coverage and sampling assumptions, without joint coverage across studies. These behavioral approximation limits are not independent evidence or task error rates. A task-oriented workflow may change the program or model or standardize presentation, then test the new target. 9 Relation to response theory and workflow analysis ANM connects response prediction with tests of representation sufficiency. Response theory supplies dynamical predictions; state and realization theories specify which distinctions a representation must preserve. Workflow analysis adds access to records and interventions. The comparison below concerns what these methods measure and what they infer from those measurements. Kubo’s response theory relates susceptibilities to dynamical correlations 23, 24. Ruelle’s review examines the response of nonequilibrium steady-state averages to perturbations of differentiable dynamics, with linear-response results under hyperbolicity and regularity assumptions 46. The finite-window tangent response used here conditions on a prepared trajectory and factors perturbation injection, propagation and readout. It requires neither an equilibrium correlation representation nor a steady-state limit. Mori–Zwanzig projection derives reduced equations with memory from specified microscopic dynamics 37, 69. ANM’s inverse experiments test whether a specified projection retains distinctions needed for the measured response; they do not infer a memory kernel or establish a closed evolution law for that projection. State-space methods clarify what a measured response identifies. Functional observability reconstructs chosen state functions from known inputs, dynamics and outputs 35, 36. ANM applies the standard linear factorization criterion to separately measured state and readout responses: a source direction with zero state response and nonzero readout response rejects the candidate. Full column rank leaves no such direction (Section 2; SI Section S1.3). Data informativity asks which properties or controllers are shared by systems consistent with observations, without necessarily identifying one system 54. Realization theory, including the Ho–Kalman construction 20, instead constructs dynamical states from input–output behavior; a finite-window response partition need not evolve recursively. Gertler’s analytical-redundancy framework constructs residuals that measure inconsistency between observed inputs and outputs and a system model, using parity equations, diagnostic observers or Kalman filters for fault detection and isolation 16. ANM’s representation tests ask whether a proposed state description can account for separately measured readout responses. A resolved violation constrains that description; it does not by itself locate a faulty component or identify a causal mechanism. State-abstraction theory classifies which model, value or policy information an aggregation preserves 25. Bisimulation metrics relate state similarity to optimal values 12, while approximate abstractions admit bounds on decision loss 1. The bound in Section 8.3 instead concerns approximation of a fixed system’s conditional output law. It follows directly from the triangle inequality; value or policy guarantees require their own dynamical and reward assumptions. Approximate information states retain reward and predictive information sufficient for approximate dynamic programming, with bounds on policy loss 50. By contrast, agreement for one preparation and readout does not by itself establish a recursive state update or a planning guarantee. For collective dynamics, El et al. 11 fit an Ising-inspired logistic law to opinions on signed networks with refreshed inboxes. The model predicts trajectories and regimes, tests unseen network families and includes a continuous-time mean-field extension. ANM instead specifies measurements of source response and tests their factorization through a candidate state. The inverse question also connects to theories that define states through future behavior. In automata theory, Nerode’s construction groups input histories that cannot be distinguished by subsequent inputs 38. Testing finitely many continuations supplies constraints on such a partition, rather than a minimal-state reconstruction for arbitrary future inputs. Stochastic bisimulation preserves rewards and block-transition probabilities for every action 17. Predictive-state representations recursively summarize action–observation histories by future-test predictions 26; causal states group histories by conditional future laws 48. Causal-State Splitting Reconstruction (CSSR) tests conditional next-symbol distributions and splits states to infer recursive predictors from discrete sequences 49. Input–output computational mechanics groups joint input–output histories by their future output laws conditional on future inputs, yielding the ϵε-transducer 4. Conditional prediction and state splitting are thus established antecedents of ANM’s representation tests. ANM tests a proposed representation under specified perturbations, readouts and finite observation windows; inference of a minimal recursive predictor is a separate objective. For the constructed program, Section 5.2 derives the coarsest sufficient partition for a declared continuation class; the other tests do not establish predictive minimality. Causal abstraction additionally requires agreement of interventional solutions or laws under maps between states and permitted interventions 45, 15, a stronger condition than local endpoint factorization. Structural causal models likewise distinguish observational and intermediate-intervention agreement 41. In AI workflows, access to records determines which of these tests is possible. Tool-using and conversational architectures expose intermediate records 64, 47, 61; GAIA supplies endpoint tasks 32. Field memory and cognitive substrates have been proposed for retention and coordination 33, 10, while traces support error analysis 5, 58. Prompt compression already includes distributional fidelity objectives: Wingate et al. 60 train soft prompts to approximate the original prompt’s conditional output distribution using KL divergence. ANM’s matched-history test evaluates a proposed state description without substituting it into the model’s input or training a compressed prompt. Prompt compression such as LLMLingua measures token reduction against downstream performance 22. LoCoMo benchmarks long-term conversational memory 30. ANM’s matched-history comparisons test fidelity to a specified system’s response distribution, which is distinct from task performance. CheckList applies behavioral and metamorphic testing through minimum-function, invariance and directional tests 44. Thus perturbing an input and checking invariant behavior is established practice. The representation tests here group comparisons by a specified candidate state, use simultaneous uncertainty bounds, and translate resolved separations into constraints on any readout based only on that state. GRADE distinguishes execution order from observed, declared or inferred dependencies and tests prediction beyond run size. Its within-corpus degree-matched analysis establishes no edge-attachment advantage, with results dependent on probe and grouping 68. AgentGraph links relations to trace spans and combines temporal replay, reconstructed-prompt perturbations and component attribution 62. Workflow diagnosis distinguishes attribution from the effect of a repair. Who&When localizes agents and steps from annotated logs 67; AIG combines step-grounded inheritance graphs with original history 3. Both score attribution against annotations. EDGE combines observational dependencies with edges validated by repairing source spans and continuing with fixed tool outputs, conditional on successful source repair 21. DoVer restores checkpoints, edits messages/plans while preserving preceding history, and scores resumed runs by success, milestones and intervention fulfillment, permitting multiple useful repairs 29. Changing a retained record addresses a different question from comparing two unchanged histories assigned the same candidate state. Sun et al. 51 restrict execution to transformed records passed between components under a fixed constraint, separately testing preservation, forbidden actions, field restoration and verification at a fixed artifact. Xiao et al. 63 cross clean/error inputs to memory updating and question generation in four counterfactual trajectories, measuring pathway effects and interactions; likelihood probes add probability-level access, and subsequent histories evolve separately. These interventions separate state and endpoint effects without establishing arbitrary compressed-state sufficiency. Activation patching replaces an internal activation, often through an interchange intervention using a donor value, do(z=zdonor)do(z=z_ donor) 65; a clamp do(z=v)do(z=v) allows any admissible v. Attribution patching approximates replacement effects using gradients 52. Mediation analysis separates direct and indirect effects through input and mediator interventions 56; a patch effect alone does not supply that decomposition. Activation interventions can propagate contextual information across positions and layers; ANM’s workflow experiments use accessible communication records. Differences concern access, representation, intervention and response resolution. Input-order studies provide a closer comparison for the scientific-evidence experiment. SciFact separates evidence selection from claim classification 57. Input-position effects 27 and few-shot example-order effects 28 concern distinct manipulations. Stable-RAG directly studies retrieved-document permutations, using hidden-state clustering and preference training to improve answer accuracy and consistency 66. ANM organizes these established criteria around a common finite-window response question. Candidate states group histories while the downstream component’s inputs stay unchanged; resolved differences constrain any readout based on those states. The experiments test this information requirement under fixed inputs, adaptive feedback and scientific-text judgments. Common empirical laws and diagnostic or repair benefits require separate matched comparisons. 10 Discussion A representation is adequate relative to a response question: which observable, under which future inputs or interaction rules? ANM connects this question to physical response prediction through common inference rules and realization-specific dynamics. Paired measurements distinguish propagation from readout sensitivity; inverse comparisons test whether the retained information is sufficient, by resolving omitted distinctions or bounding agreement on tested histories. Physical calibration tests predictions from known delayed dynamics. The AI studies test information requirements for measured behavior, including adaptive actions that select later observations. Fidelity to a fixed downstream component includes its errors and presentation sensitivity; it is distinct from the information needed for correct task performance. In SciFact, preserving the measured response can require distinguishing document orderings, whereas a task-oriented design may standardize presentation or change the reader. Agreement can also coexist with judgments contrary to source annotations. These comparisons characterize observable behavior. Inference about internal representations or reasoning requires additional evidence; a changed target requires a new comparison. Prompt compression selects information for later inference 22; long-conversation benchmarks evaluate whether a model retains information needed for later tasks 30. Checkpointing, caching and passing records between components likewise involve choosing which information remains available. Matched-history tests assess a proposed summary while the model still receives complete inputs. They identify distinctions the summary must preserve; new comparisons then test a revision. Evaluating actual input compression additionally requires running the model on the compressed inputs. A cache reproducing the reader’s output distribution, for example, cannot merge the rejected SciFact orderings at the tested tolerance. Any distribution using only their unordered excerpt description incurs TV error at least 0.37500.3750 on one input, under the study’s simultaneous confidence guarantee (Eq. (32)). This constrains behavioral approximation, without prescribing a unique encoding or task error rate. Support on one set of histories does not automatically extend to another. The association state is supported on four histories for recipient A, a restriction chosen after broader development. Independent identifier/order changes leave three pairs unresolved under the fixed bounds; their categorical separations are exploratory. A later comparison uses new inputs without resampling that anomaly and supports both the simple state and its revision, so added detail has no demonstrated benefit (Sections 5.4–5.5). The two-decision workflow tests a prescribed requirement: instructions give current-version assessment, the decision rule and identifier/order invariance, and the experiment program joins the records. The two coarse summaries are constant, so their rejections express one missing-test distinction. The instructed rule predicts every action; fitted forecasts quantify the cost of that omission for those predictors, without method advantage or optimal-compression claims. A supplementary final-action comparison shows why readout choice matters: a summary may preserve release decisions while omitting which tests precede them. Unresolved adequacy can reflect the comparison design rather than precision (Table 2). The graph experiment’s full-column-rank state-contrast matrix leaves no nonzero source contrast with zero state change; more repeats cannot create one. New source directions or matched-state histories are needed. Tests across systems likewise evaluate each system separately: a shared representation permits distinct output laws, without pooling them into an artificial distribution (Section 8). Representation information and intervention benefit. Changing a workflow asks a different question from testing a description of its behavior. In an exploratory inventory task, explicitly joining requests to returns restores a required quantity/allocation contrast on one known case. A direct check of request–return associations selects the same operation, and fixed joining costs fewer calls. These exploratory comparisons followed failed calibration; scheduled validation was not performed (SI Section S7.3). In a prospective four-program pilot, replacing one integral corrects that output while both faulty programs retain incorrect target times or coordinates (SI Section S7.2.2). A SciFact development comparison likewise separates response measurement from benefit. On 89 tasks, the response-guided policy and direct reading both reach 74.72% correctness, at 7.56 and 1.00 calls per task; ordinary features give 75.28% at 2.55 calls without response measurements. Initial evidence selection is invalid or unavailable on 45 tasks, limiting measurement coverage without explaining the absent gain (SI Section S7.4). Establishing intervention benefit requires independent tasks, fixed policies, comparators with the same information, and measurement and execution costs. These development estimates establish no policy superiority. A separate constructed software-build workflow uses the same Qwen model to test versioned job and dependency records. Wrong-component, obsolete-revision and obsolete-build histories produced READY on all 24 calls; across all eight faulty histories, none of 32 calls produced a valid HOLD. Overall correctness is 32/64, equal to always answering READY. Its preliminary representation comparison selected no candidate, so the planned confirmation was not run. These observations delimit that task and interface. They are not an ablation of the closed-loop interface: task, presentation and output requirements also differ, so the cause of the performance difference is not isolated (SI Section S7.5). Scope. The AI studies concern accessible workflow records and analyst-selected states. Closed-loop support concerns one model, fixed version names during confirmation, and two decisions from a specified preparation; the tested summary omits pending-request information needed at later times. It establishes neither autonomous reduced dynamics nor a numerical discretization of the continuous field law. Simulated tool checks also delimit transfer to live software systems. Interface development failures and feedback pilots without task benefit remain separately reported (SI Section S5). Tests of revised representations require fixed downstream components and interfaces, informative comparisons and independent error controls. Arbitrary-history sufficiency, causal mediation, autonomous reduction, cross-model transfer, jointly formed multi-agent states, emergent transitions and common empirical response laws require separate evidence. 11 Measurement and reconstruction conventions Each comparison fixes its source coordinates, candidate representation, readout, observation window and external conditions. Fixed-input studies hold the subsequent input sequence constant. Adaptive studies hold the action policy, tool-response rule and observation intervention constant, allowing actions to select later observations. Preparation is fixed except for the specified source or history contrast. A representation computed only for analysis does not alter the downstream component’s input; a presentation change is instead a system intervention. These access distinctions are stated for each experiment in Table 1 and the corresponding experimental sections. Additional intervention protocols appear in SI Sections S7.3 and S7.4. Experimental units are simulation seeds, source directions, program cases or scientific claims, as specified in each result. Repeated calls and comparisons that reuse endpoints are not independent tasks. Missing or malformed responses remain in the accounting, with dependent measurements marked unavailable rather than replaced by zero. The SI describes the available sampling, parsing and scoring rules, uncertainty calculations and controls, and identifies gaps in historical specifications. Numerical reanalysis recomputes statistics from recorded observations; reexecution reruns the program or graph. Replication requires new model calls, and authentication of historical execution requires separate provenance evidence. SI Table S6 identifies each study’s evidential status and limits. The described methods permit independent implementation; the absence of public raw records and executable code prevents independent reconstruction of all reported numerical results from the publication alone. Acknowledgements. We thank Herbert Levine, Eduardo Sontag, Jing Wang and Antong Chen for helpful discussions. Generative AI tools assisted research design, coding, analysis, literature review, figure preparation and manuscript editing. The authors take full responsibility for the work. Data availability. The study uses simulated physical data, constructed AI tasks and recorded model outputs. Public sources comprise SciFact training claims and scientific abstracts; historical comparisons use author-constructed variants of GAIA tasks. The manuscript and SI specify the data sources, task construction, measurement rules and reported results. Raw experimental records have not been publicly deposited; SI Section S8 states the resulting reconstruction limits. Code availability. 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Equations, figures and tables use an S prefix; references to the main manuscript are identified explicitly. Contents 1 Introduction 2 A common response framework for physical and AI systems 3 Forward response in physical systems 4 Forward response in an AI event graph 5 Inverse tests under fixed future inputs 6 Inverse tests under tool feedback 7 Inverse tests with scientific text 8 Information required to preserve responses 9 Relation to response theory and workflow analysis 10 Discussion 11 Measurement and reconstruction conventions References Supplementary Material S1 Finite-window response and representation adequacy S2 Field dynamics and conditional response S3 Physical measurement methods and additional results S4 Methods for inverse tests under fixed inputs S5 Methods for inverse tests under tool feedback S6 Methods for scientific-text comparisons S7 Graph response methods and additional workflow studies S8 Evidence status, data and code availability S9 Notation and terminology References S1 Finite-window response and representation adequacy S1.1 States, sources and readouts Forward response predicts how a source change affects an observable; inverse comparison tests whether a candidate state preserves the distinctions needed for that response. The definitions below separate readout adequacy from autonomous dynamics. The physical specialization in Section S2.1 adds delay histories and well-posedness conditions; stationary susceptibility also requires closure and stability. Throughout, an admissible comparison fixes the instance class, interface, loading coordinate, stopping rule and readouts. State coordinates and dynamics are realization-specific. For a linear map, ImIm is its set of possible outputs and ker its set of inputs mapped to zero. In response formulas, Δ denotes finite contrasts and δ tangent perturbations; hats mark measured or finite approximations. The derivative DxFD_xF maps a perturbation of x to the first-order change in F: a Jacobian in finite dimensions, or a Fréchet derivative on the specified function space. Distributed-system apparatus and experimental instance. A distributed-system apparatus supplies external memory, tools, evaluation and execution control around local operators. An instance a specifies one task or experiment under fixed interfaces, permissions and stopping rules. It labels the preparation, not a dynamical coordinate. Retained state. For family f, let XfX_f be the full state space, including the variables, required histories, observation records and directly consumed inputs of that realization, with st∈Xfs_t∈ X_f. Fields and local memories are physical examples. Let ZfZ_f be a candidate coordinate space. A declared map Gf:Xf→ZfG_f:X_f→ Z_f gives zt=Gf(st)z_t=G_f(s_t), the retained state. This coarse-graining selects information from the full state. Workflow coordinates may encode unresolved tasks, conflicting or delayed evidence and pending corrections. Adequacy for a readout is tested separately; autonomous closure is an additional condition. Preparation, coarse-graining, and endpoint readout maps. Preparation Pfprep(a)=s0P_f prep(a)=s_0 sets the initial full state; dynamics and inputs give sT∈Xfs_T∈ X_f. The candidate is zT=Gf(sT)∈Zfz_T=G_f(s_T)∈ Z_f. The endpoint readout OT:Xf→fO_T:X_f _f takes values in a declared observation space; numerical observables may be scalar or vector valued. A readout depending only on the retained state would be a map O¯T:Zf→f O_T:Z_f _f with OT(s)=O¯T(Gf(s))O_T(s)= O_T(G_f(s)). The bar distinguishes its domain; it is not an average. This composition, also written OT=O¯T∘GfO_T= O_T G_f, is a condition to test. The nested case uses outer workability readout PfP_f and contained endpoint readout QfQ_f, with w=Pfw=P_f, r=Qf/Pfr=Q_f/P_f for Pf>0P_f>0 (main-text Section 3.3); these differ from preparation. S1.2 Window observations as terminal readouts Trajectory measurements become terminal readouts by retaining their records in sTs_T, together with any directly consumed input. For a measured field component ctc_t of sts_t, an integral can be recorded by appending ˙t=ψ(ct,t) a_t=ψ(c_t,t), 0=0a_0=0, so T=∫0Tψ(ct,t)ta_T= _0^Tψ(c_t,t)\,dt. The weight ψ selects the window and real/imaginary lock-in components. Amplitude and phase are functions of these components; differentiable phase requires a local continuous branch at nonzero amplitude. Other path readouts retain their corresponding records. These passive observations neither feed back into the dynamics nor establish candidate adequacy or autonomous stability. S1.3 A falsifiable retained-state criterion A candidate state misses a response-relevant distinction if a source change leaves that state unchanged but changes the endpoint. The first-order test compares the source-to-state and source-to-endpoint maps on the same measured perturbations; it does not assume a state-only endpoint law in advance. At a declared operating point, let TU_T be the finite-dimensional measured perturbation space and T,TZ_T,Y_T the vector spaces of retained-state and endpoint variations. Define the first-order maps z,T:T→T,O,T:T→TK_z,T:U_T _T, _O,T:U_T _T from sources to states and endpoints, respectively. On the declared time grid, let utu_t denote the controlled source input and let the full-state transition be st=Ft(st−1,ut)s_t=F_t(s_t-1,u_t), with At=DsFtA_t=D_sF_t and Bt=DuFtB_t=D_uF_t along the baseline trajectory. Define Φ(T,t)=AT⋯At+1 (T,t)=A_T·s A_t+1 and Φ(T,T)=I (T,T)=I, the identity map. Repeated substitution in its tangent recursion gives δsT=Φ(T,0)δs0+∑t=1TΦ(T,t)Btδut.δ s_T= (T,0)δ s_0+ _t=1^T (T,t)B_tδ u_t. Applying DsGfD_sG_f and DsOTD_sO_T gives the two response maps without assuming factorization. Allowed preparation changes join source changes in TU_T; fixed preparation sets δs0=0δ s_0=0, as in main-text Eq. (3). First-order retained-state criterion. There is a unique linear readout map, with superscript (1)(1) denoting first order, ℋT(1):Imz,T→TH_T^(1):ImK_z,T _T satisfying O,T=ℋT(1)z,TK_O,T=H_T^(1)K_z,T on the measured perturbation space if and only if kerz,T⊆kerO,T. _z,T _O,T. Uniqueness is only on the state displacements reached by the measured perturbations; the result makes no claim about unprobed state directions. Derivation. If the factorization holds, every perturbation that leaves the retained state unchanged also leaves the endpoint unchanged. Conversely, under the kernel inclusion define ℋT(1)(z,Tδu):=O,TδuH_T^(1)(K_z,Tδ u):=K_O,Tδ u. Two perturbations producing the same state displacement differ by a vector in kerz,T _z,T; the inclusion makes their endpoint displacements equal. The resulting map is therefore well-defined and linear, and fixes ℋT(1)H_T^(1) uniquely on Imz,TImK_z,T. The delayed-history example in main-text Section 3 gives a present-state null direction with a nonzero later readout response. Equal current values do not mean equal initial histories, so solution uniqueness is preserved. The violation requires a revised description to separate those histories without prescribing a unique encoding. The criterion remains local and first order, rather than an exact nonlinear factorization theorem. Let T(u) z_T(u) and T(u) o_T(u) denote the nonlinear retained-state and endpoint maps from a source history u. Exact sufficiency on a finite neighborhood instead requires that any two source histories producing the same retained state also produce the same endpoint: T(u1)=T(u2)⟹T(u1)=T(u2). z_T(u_1)= z_T(u_2) o_T(u_1)= o_T(u_2). For continuously differentiable (C1C^1) maps between finite-dimensional coordinate spaces, constant rank of T z_T and kernel inclusion throughout a neighborhood imply factorization on a sufficiently small local neighborhood. To see why, use constant-rank coordinates (x,y)(x,y) in which T z_T depends only on x and distinguishes nearby x values. The y directions lie in kerDT D z_T, so kernel inclusion gives DyT=0D_y o_T=0. On a sufficiently small product neighborhood, T o_T is independent of y and is a function of T z_T. Without the constant-rank condition, pointwise kernel inclusion alone is insufficient: (u)=u2 z(u)=u^2 and (u)=u3 o(u)=u^3 satisfy the derivative-kernel condition at every scalar u, yet the arbitrarily close inputs u=ϵu=ε and u=−ϵu=-ε have the same retained state and different endpoints. Interpreting measured contrasts. A resolved same-state violation can reject adequacy without a separate positive control. Informative empirical local support covers only tested comparisons and requires nontrivial null directions or matched histories, a detectable endpoint positive control, adequate comparison coverage, specified equivalence bounds and controlled matching error. These support conditions are not necessary for mathematical sufficiency: a constant readout factors through a constant state. Fix source intervention, units, window and endpoint rule; estimate state and endpoint contrasts and their resolution separately. Joint changes establish association under intervention, not mediation; state changes without endpoint changes can reflect cancellation or robustness. Unresolved responses do not localize effects. For an approximate match with a deterministic readout, suppose the true state difference is bounded by εz _z, the error in the measured endpoint contrast ΔO O by εO _O, and candidate readouts O=h(z)O=h(z) have a Lipschitz bound LhL_h on the compared states justified independently of the tested endpoint contrasts. Factorization then requires ‖ΔO^‖≤Lhεz+εO\| O\|≤ L_h _z+ _O. Exceeding the bound rejects this Lipschitz readout class. Exact matching sets εz=0 _z=0; approximate matching cannot reject arbitrary readouts without independent sensitivity control. Stochastic laws need a specified comparison and uncertainty bound. For example, z=uz=u, O=2uO=2u preserves the input but rejects Lh≤1L_h≤ 1 on every distinct pair when εz=|Δu| _z=| u|, εO=0 _O=0. This rejects the sensitivity constraint, not the state; check the readout assumptions and matching before refining the representation. The violation cannot itself justify a replacement bound. Inverse use: distinctions a representation must preserve. Response equivalence groups histories whose specified responses agree under every common continuation, fixing sources, readouts, windows and noise laws. Compare conditional laws for distributional claims, means for mean claims, and baseline levels when levels rather than contrasts are targeted. The coarsest sufficient partition constrains refinement without supplying a reconstruction algorithm or autonomous realization. Related constructions include causal states and predictive-state representations 48, 26 (main-text Section 9). Finite tests constrain only their comparisons; tolerance-based matching need not be transitive. In the workflow examples, the downstream component is the program or language model that reads the records and produces the output. Main-text Section 8.1 gives the finite-history shared-representation criterion and its measurability argument; main-text Eq. (29) gives the transfer bound. Both concern declared systems and continuations. Transfer additionally requires an independently justified uniform bound on the change of readout law, which finite measurements alone do not supply. State measurements and selective interventions. This deterministic example asks which measurement or selective intervention distinguishes altered propagation from altered readout. In fixed physical units, let gpg_p be the propagation gain and grg_r the readout gain. Assume z=gpu,O=grz,z=g_pu, O=g_rz, with zero baseline, no direct source–endpoint path, and calibrated healthy gains gp=gr=1g_p=g_r=1. The two changes (gp,gr)=(1/2,1)(g_p,g_r)=(1/2,1) and (1,1/2)(1,1/2) have the same complete source–endpoint map O=u/2O=u/2. Before state-clamp or repair feedback, source interventions observed only at the endpoint cannot distinguish them. State measurements do distinguish z=u/2z=u/2 from z=uz=u. Alternatively, a known selective clamp, written do(z=v)do(z=v), sets the state to v while holding u fixed. It gives O=grvO=g_rv and identifies the readout gain without separately measuring the natural state. Estimate gpg_p from source–state contrasts and grg_r from independent clamps, then compare predicted and measured endpoints and calibrated gains. With one reset allowed, reset the sole changed gain, leave healthy systems alone, and abstain when both change. At u=1u=1, the correct reset takes O=1/2O=1/2 to 11, while the other leaves O=1/2O=1/2. These are equation-based deductions; ordinary state-space identification with the same information gives the same decisions. The assumptions matter: z=uz=u, O=2zO=2z and z=uz=u, O=2uO=2u give identical source/state/endpoint records although only the first endpoint depends on the state. A selective state clamp separates them. A fit to source interventions alone therefore cannot certify mediation. Outside the declared class, a mismatch calls for competing mechanisms rather than a forced component label. Decision value also depends on the readout. For δz=diag(1,4)δuδ z=diag(1,4)δ u and δO=(1/4,1/100)δzδ O=(1/4,1/100)δ z, positive unit source steps have state norms 1,41,4 but endpoint changes 1/4,1/251/4,1/25. For a fixed upper-boundary gap 1/41/4, equal step costs and allowed steps of at most one, only the first axis reaches the boundary. Standard constrained control gives the same choice. Exact-arithmetic implementation checks verify these deductions, including zero-gain, healthy, multiple-change and direct-path controls. These are analytic checks, not measured AI fault or repair results. S1.4 Source encoding and response equivalence When a workflow encodes evidence as numerical sources, an explicit map connects visible records to source locations and amplitudes. Its specification identifies the visible inputs, event fields, preparation and admission rules, and assignment of numerical sources. The definition below separates these operations; main Section 4 gives the implemented graph construction, state update and readout together, and Section S7.1.1 gives its sampling procedure. Encoding records as a graph is one realization: a language model may instead read records directly. Evidence-to-field source map. Let Ef≤tE_f^≤ t be the set of evidence histories up to time t. For a history η∈Ef≤tη∈ E_f^≤ t, proposed, prepared and admitted records efprop,efprep,efadme_f prop,e_f prep,e_f adm form the source chain η→ξfefprop→Afprepefprep→Afadmefadm→Jfjf(η)∈Yfsrc.η _fe_f prop A_f prepe_f prep A_f adme_f adm J_fj_f(η)∈ Y_f src. Here YfsrcY_f src is the source-injection space. Extraction ξf _f proposes events; preparation AfprepA_f prep preserves evidence visible before verification, applies prespecified support-amplitude bounds based on that evidence, and excludes endpoint labels. Admission AfadmA_f adm selects events allowed to act. Finally JfJ_f assigns numerical amplitudes, sites, channels and operator tags. The composed source map is jf=Jf∘Afadm∘Afprep∘ξfj_f=J_f A_f adm A_f prep _f, with the stages applied in the order shown above. The evidence process, action set, event alphabet, relation types and verifier are family-specific; the source-chain structure applies to simulations, recorded data and model interfaces. S1.4.1 When the source-event representation is sufficient A continuation fixes later inputs or external response and observation rules, and the readout time, while preserving interface, dynamics and stopping rules. For an adaptive system it fixes the policy rather than its sampled actions (main-text Section 2.1). Source sufficiency asks whether the event representation preserves the observable effects of evidence histories under every allowed continuation. Source-representation sufficiency. Fix an admissible neighborhood, a common set of admissible continuations, a horizon T, and the scalar or vector readouts to be compared. Include verifier outputs or operating criteria among these readouts when required. Let ρf:Ef≤t→Rfrep _f:E_f^≤ t→ R_f rep map evidence histories to a candidate representation space RfrepR_f rep. Define η1≡fη2 _1 _f _2 when replacing jf(η1)j_f( _1) by jf(η2)j_f( _2) in every common admissible continuation leaves the declared readouts unchanged. For deterministic readouts, this means equality after the same fixed calibration or quantization map. For stochastic readouts, compare the declared conditional statistic or readout distribution given each evidence history under the same continuation and noise law. This defines an equivalence relation, rather than a pairwise tolerance test. In the nested specialization, the readouts include Pf,QfP_f,Q_f, from which w=Pfw=P_f and r=Qf/Pfr=Q_f/P_f are derived where Pf>0P_f>0. Then ρf _f is sufficient for the local source representation if and only if no two inequivalent histories are assigned the same representation, ρf(η1)=ρf(η2)⟹η1≡fη2. _f( _1)= _f( _2) _1 _f _2. The coarsest sufficient representation is obtained when each representation class is exactly one equivalence class of ≡f _f. This is an abstract quotient; a finite-valued representation exists only when there are finitely many classes. The quotient argument alone supplies no finite-dimensional regular or autonomous realization. Derivation. If two inequivalent evidence histories are assigned the same representation, then any source construction that factors through that representation assigns them the same numerical source term although some admissible continuation separates them through a declared readout. Such a representation cannot reproduce all specified continuation readouts. Conversely, if each representation class contains only histories equivalent under ≡f _f, then, for each fixed admissible continuation, its calibrated deterministic values or declared conditional statistics or laws are functions of the representation. This establishes readout sufficiency on those tests; autonomous evolution of the representation requires a separate closure condition. Counting failed checks can discard what each failure concerns. A rejection of a relation in use and a delayed warning about an unused candidate may have the same count but different source locations, signs or delay channels in YfsrcY_f src. The count is insufficient only if an allowed continuation makes their declared Pf,QfP_f,Q_f or OTO_T readouts differ. A revision must retain the locations, channels or operator distinctions needed to separate such responses. Differences may be discarded only when every declared continuation leaves them invisible to the readout. S1.5 Comparing realization-specific retained responses Comparing realizations requires a state correspondence and an endpoint map; nested-readout agreement alone need not preserve other observables. Retained response and endpoint readout. Let ireach⊂ZfiZ_i^reach⊂ Z_f_i, i=1,2i=1,2, be the retained states reached in two realizations, and let J:1reach→2reachJ:Z_1^reach _2^reach be a specified comparison map. Assume each endpoint factors through its retained coordinate, with maps O¯fi,T O_f_i,T into a common declared readout space. Exact endpoint values agree on corresponding states z2=J(z1)z_2=J(z_1) if and only if O¯f2,T∘J=O¯f1,T O_f_2,T J= O_f_1,T there. If the reached sets are locally smooth and these maps are differentiable, first-order endpoint changes under δz2=DJ(z1)δz1δ z_2=DJ(z_1)δ z_1 agree if and only if DO¯f2,T(J(z1))DJ(z1)δz1=DO¯f1,T(z1)δz1D O_f_2,T(J(z_1))\,DJ(z_1)δ z_1=D O_f_1,T(z_1)δ z_1 for every declared reached tangent direction δz1δ z_1. Parentheses specify where each derivative is evaluated; J need not be linear. For the optional nested pair, if (Pf2,Qf2)∘J=(Pf1,Qf1)(P_f_2,Q_f_2) J=(P_f_1,Q_f_1), corresponding states have the same reduced readouts (w,r)(w,r) for every z1∈1reachz_1 _1^reach with Pf1(z1)>0P_f_1(z_1)>0. Derivation. Exact endpoint agreement is the composition identity for O¯fi,T O_f_i,T; the derivative condition follows by the chain rule on corresponding displacements. Exact composition implies this condition; equal derivatives alone do not imply equal endpoint values, since the maps may differ by an additive constant. For scalar endpoint values, agreement only of their declared ordering is weaker than equality of the values. The nested-pair statement follows by composition with J and division by the common positive outer readout. Interpretation. Preparation, dynamics and readout projection remain realization-specific. No comparison map J is assumed between the reported physics and AI systems. A comparison requires specified state correspondence or matched readout semantics; different projections can resolve different changes. S1.6 Minimal representation of a terminal response For fixed sources, terminal readout and finite horizon, linear response identifies a minimal representation of the reached displacements visible to that readout. This differs from independently declaring a physical retained state. Consider a time-independent closed linear realization with retained-state transition A, source injection B, and differentiably factored terminal readout OT=O¯T∘GfO_T= O_T G_f. For δzt=Aδzt−1+Bδutδ z_t=A\,δ z_t-1+B\,δ u_t with zero initial displacement, write CT=DzO¯TC_T=D_z O_T. Let ℛTR_T contain terminal displacements reachable by the allowed sources, and TN_T contain displacements invisible to the terminal readout: ℛT=Im[B,AB,…,AT−1B],T=kerCT.R_T=Im[B,AB,…,A^T-1B], _T= C_T. For time-dependent transitions, ℛTR_T is instead the image of the finite-window source-to-state map. The quotient, which identifies displacements differing by an invisible component, ℛT/(ℛT∩T)R_T/(R_T _T) is isomorphic to CTℛTC_TR_T: it retains exactly the terminal displacements that this readout distinguishes. The discarded component may matter for another readout or later time. This quotient need not evolve autonomously. For a finite-dimensional, time-independent linear input–output system, a minimal dynamical realization is reachable and observable. Constructing a dynamical quotient uses transition-invariant reachable and unobservable subspaces; the kernel of one terminal readout need not be invariant. An effective dimension may instead summarize singular-value concentration. Let σk _k be the singular values of a nonzero finite-window response matrix at a stated horizon. For a time-independent closed transition with spectral radius ρ(A)<1ρ(A)<1, the stationary choice is χ∗G=(I−A)−1B _ ^G=(I-A)^-1B, derived in Section S2.2. For the modal energies σk2 _k^2, the participation ratio is rPR=(∑kσk2)2∑kσk4r_ PR= ( _k _k^2 )^2 _k _k^4 Endpoint projection by DzO¯TD_z O_T can reduce rank; an injective non-isometric readout can still change the singular-value spectrum. Terminal minimality is relative to the declared source and readout. The constructed code-repair graph in Section S7.1 retains 3232 coordinates. Identity observation under a constructed full source basis does not establish empirical minimality for the endpoint pair O=(Pf,Qf)O=(P_f,Q_f). S2 Field dynamics and conditional response S2.1 Formal delayed field-coupled model The field model realizes finite-window response through medium propagation, local sources and prescribed delayed inputs. Their distinction determines which histories enter the dynamical state. Let H=L2(X,ℝq)H=L^2(X,R^q), or another separable Hilbert field space, with X a spatial or graph domain and q realization-specific channels. Here L2L^2 is the space of square-integrable fields. The shared field is ct∈Hc_t∈ H. Local operator i, among N operators, has memory mi(t)∈Mim_i(t)∈ M_i in a finite-dimensional or Banach space and contributes Ji(ct,mi(t))∈HJ_i(c_t,m_i(t))∈ H. Physical channels may encode activity; workflow channels may encode candidate weights, conflicting evidence, inhibition or tool availability. The field model is ∂tct=Lct+∑i=1NJi(ct,mi(t))+∑ℓ=1NτEℓyℓ(t−τℓ), _tc_t=Lc_t+ _i=1^NJ_i(c_t,m_i(t))+ _ =1^N_τE_ y_ (t- _ ), (S2.1) Here L generates autonomous medium evolution. The NτN_τ channels supply prescribed histories yℓ(t−τℓ)∈Yℓy_ (t- _ )∈ Y_ , whose generators lie outside the modeled operators, through bounded couplings Eℓ:Yℓ→HE_ :Y_ → H. Internal delayed feedback belongs to JiJ_i, including the consensus coupling. The displayed local-source arguments suppress required internal histories; the augmented state below includes them explicitly. For instantaneous reads, a field-coupled local operator comprises a read operator Ri:H×Mi→OiR_i:H× M_i→ O_i, a local update map πi:Oi→Ui _i:O_i→ U_i, a write operator Wi:Ui→YisrcW_i:U_i→ Y_i src, a source-coupling operator Gi:Yisrc→HG_i:Y_i src→ H, and a memory-update rule Ψi _i. Its source contribution is Ji(c,mi)=GiWi(πi(Ri(c,mi))).J_i(c,m_i)=G_iW_i\! ( _i(R_i(c,m_i)) ). Here OiO_i is the local readout space, UiU_i the update-output space, and YisrcY_i src the source space exposed by operator i. Assume L generates a strongly continuous contraction semigroup on H; spatial and graph examples appear in Table S1, with diffusion coefficient or tensor D, relaxation operator Γ and graph Laplacian LGL_G. For a PDE, the local write can enter through an injection kernel, (Giw)(x)=∫Xgi(x,ξ)w(ξ)ξ,(G_iw)(x)= _Xg_i(x,ξ)w(ξ)\,dξ, where w denotes the written source density in this formula. On a finite graph, GiG_i selects the sites and channels receiving the write. The injection kernel gig_i places a write into the field; the Green kernel KLK_L of ∂t−L _t-L propagates it. When the propagator admits a kernel, interpreted weakly when necessary, a specified source contribution is δc(t,x)=∫0t∫XKL(t−s,x,x′)(Giδws)(x′)dx′s,δ c(t,x)= _0^t\! _XK_L(t-s;x,x )\,(G_iδ w_s)(x )\,dx \,ds, with the full coupled response additionally including derivatives of local sources, memory dynamics and internal feedback, incorporated in the augmented state below. An external delayed coupling has finite gain βℓ _ : ‖Eℓy‖H≤βℓ‖y‖\|E_ y\|_H≤ _ \|y\| on the relevant signal space. For ‖yℓ‖≤Mℓ\|y_ \|≤ M_ on the sampled history, forcing is bounded by βℓMℓ _ M_ and its integrated squared norm by Tβℓ2Mℓ2T _ ^2M_ ^2. A bounded coupling alone does not bound arbitrary inputs or long-time state growth. Table S1 summarizes the operators. Table S1: Field operators. Sources resolve as Ji=GiWi(πi(Ri))J_i=G_iW_i( _i(R_i)). Here D, Γ and LGL_G denote diffusion, relaxation and the graph Laplacian. Consensus delay is internal; EℓE_ couples externally generated records, even when an operator action caused them. Symbol Physical meaning Typical realization L autonomous propagation, relaxation, and dissipation of the shared medium diffusion–decay operator DΔ−ΓD - , graph Laplacian generator −DLG−Γ-DL_G- , or Markov/contraction generator JiJ_i composite source generated by modeled local operator i feedback force, actuator drive, graph-node source, or typed evidence increment RiR_i local readout available to operator i from the field and memory projection, local averaging, attention window, tool-state readout πi _i local update or feedback law used by operator i controller law, consensus update, model inference, decoding, or tool choice WiW_i local write produced by operator i’s update rule signed source density, graph-node injection vector, channel-specific evidence update GiG_i source-coupling operator placing the local write into the medium spatial source kernel gi(x,ξ)g_i(x,ξ) or finite source matrix on a graph EℓE_ finite-gain coupling from delayed input outside the modeled local operators into the medium verifier, tool, environment, or measurement channel with delay τℓ _ Continuous memory dynamics are specified by m˙i(t)=Ψi(ct,mi(t),ηi(t)),i=1,…,N, m_i(t)= _i(c_t,m_i(t), _i(t)), i=1,…,N, (S2.2) where ηi(t) _i(t) is bounded apparatus input. The well-posedness result below applies under its continuous-time hypotheses. Discrete or hybrid memory needs its own recurrence or jump law. Externally recorded memory is instead a prescribed input, subject to source regularity. Equivalent source terms. Fix the same initial field, the same external inputs and all other field-source histories. If two local mechanisms expose identical effective source histories, Ji(ct,mi(t))=Jj(ct,mj(t))J_i(c_t,m_i(t))=J_j(c_t,m_j(t)) for almost every t∈[0,T]t∈[0,T], variation of constants gives the same field trajectory on that horizon. Every declared readout depending only on this field trajectory therefore agrees. For readouts that also use separately retained memories or other coordinates, equal additional inputs are sufficient for agreement; otherwise the endpoints must be compared through the full readout map. Resolved component maps permit selective interventions, but equal exposed sources do not identify their generators. Component diagnosis needs additional measurements or interventions; workflow retriever, verifier and tool outputs supply source or input channels. S2.2 History, projection and conditional autonomous response Projection preserves selected information; autonomous evolution requires closure. Delayed dynamics must first be described on a state containing their internal histories. Include internal history before projecting. For each of the NintN_ int internally generated delayed quantities qaq_a, retain ξat(θ)=qa(t+θ) _a^t(θ)=q_a(t+θ) on [−τa,0][- _a,0]. Together with the field and local memories, these histories form s(t)=(ct,m1(t),…,mN(t),ξ1t,…,ξNintt)∈Xf.s(t)=(c_t,m_1(t),…,m_N(t), _1^t,…, _N_ int^t)∈ X_f. Each qaq_a must be a specified causal function of modeled variables; any additional signal generator needs its own state equation. Internal histories evolve by ∂tξat=∂θξat _t _a^t= _θ _a^t, with boundary value ξat(0)=qa(s(t)) _a^t(0)=q_a(s(t)). Here u denotes the control parameters. Field evolution, memory laws m˙i=Ψi(c,mi,ξ,u) m_i= _i(c,m_i,ξ;u), and these history shifts define s˙=Ffull(s,u,yt). s=F_ full(s;u,y^t). (S2.3) External histories yty^t remain prescribed inputs. The history space for mild solutions is specified in Section S2.3; these derivatives require classical solutions in the generator domain. Append observation records when needed (Section S1.2). Declare z=Gf(s)∈Zfz=G_f(s)∈ Z_f before testing factorization: the current field can omit relevant history, and endpoint factorization alone does not establish autonomous closure. Closure on an invariant branch. Fix control u and stationary external input y¯ y. Suppose a locally invariant branch admits a continuously differentiable (C1C^1) parametrization s=hu(z)s=h_u(z) with Gf∘hu=idG_f h_u=id, where GfG_f is also C1C^1. Branch trajectories must be classical solutions: fields lie in the operator domain D(L)D(L), and histories have compatible derivatives and boundary traces. Assume Ffull(hu(z),u,y¯)F_ full(h_u(z);u, y) is tangent to this branch. Then DsGfDzhu=ID_sG_f\,D_zh_u=I, and the chain rule gives z˙=DsGf(hu(z))Ffull(hu(z),u,y¯)=:fu(z). z=D_sG_f(h_u(z))F_ full(h_u(z);u, y)=:f_u(z). (S2.4) Exact closure holds for trajectories initialized on the branch at fixed u. Changing control can move the branch away from the same full-state preparation, so f(z,u)=fu(z)f(z,u)=f_u(z) alone does not establish controlled transients. A sufficient extension is DsGf(s)Ffull(s,u,y¯)=f(Gf(s),u),s∈,D_sG_f(s)F_ full(s;u, y)=f(G_f(s),u), s , for every admitted control, on a common domain ⊂XfD⊂ X_f containing the preparations and invariant under admitted controlled evolution, with classical solutions. Without exact closure, a reduced transient approximation requires initialization and tracking error bounds, or the missing variables must remain in the response state. Attraction alone does not eliminate transients from initialization away from the branch. Arbitrary external forcing requires a controlled description or its generator in s. Finite-window response requires neither branch reduction nor a potential representation. Stable equilibrium and discrete response. Equilibrium susceptibility follows from stationary balance; identifying it with accumulated transient response requires additional stability and closure conditions. For a finite-dimensional branch family f(z,u)=fu(z)f(z,u)=f_u(z) that is C1C^1 jointly in state and control on a common coordinate neighborhood, let z∗z_ be a stationary point at u∗u_ . The linearized reduced equation is δz˙=Azδz+Bzδu,Az=Dzf(z∗,u∗),Bz=Duf(z∗,u∗),δ z=A_zδ z+B_zδ u, A_z=D_zf(z_ ,u_ ), B_z=D_uf(z_ ,u_ ), (S2.5) If AzA_z is invertible, the implicit-function theorem gives a local stationary branch, and differentiating f(z∗(u),u)=0f(z_ (u),u)=0 yields δz∗=χ∗δu,χ∗=−Az−1Bz.δ z_ = _ δ u, _ =-A_z^-1B_z. (S2.6) Tangency makes hu(z∗(u))h_u(z_ (u)) a full-system equilibrium; this stationary susceptibility needs no controlled transient closure. If AzA_z is also Hurwitz (all eigenvalues have negative real parts), the equilibrium is locally attracting within its fixed-control branch. For the linearized reduced equation, a constant source and zero displacement give the integral of eAztBze^A_ztB_z, converging to −Az−1Bz-A_z^-1B_z. Under the controlled closure above, this also gives the full system’s retained-state linear response; an approximation instead requires an explicit response-error bound. A moving equilibrium adds −χ∗(u)u˙- _ (u) u to its deviation equation and requires a separate tracking bound. For a verified closed retained-state transition, with a differentiably factored readout O¯T O_T, let AtretA_t ret and BtretB_t ret be its Jacobians with respect to the retained state and source input along the reference trajectory. Write δzt δ z_t =Atretδzt−1+Btretδut, =A_t retδ z_t-1+B_t retδ u_t, (S2.7) Φret(T,t) _ ret(T,t) =ATret⋯At+1ret, =A_T ret·s A_t+1 ret, δOT δ O_T =DzO¯T[Φret(T,0)δz0+∑t=1TΦret(T,t)Btretδut]. =D_z O_T [ _ ret(T,0)δ z_0+ _t=1^T _ ret(T,t)B_t retδ u_t ]. Here Φret(T,T)=I _ ret(T,T)=I; changing initial history can make δz0≠0δ z_0≠ 0. For time-independent A,BA,B with spectral radius ρ(A)<1ρ(A)<1 (the spectrum lies inside the unit circle), a persistent tangent source has the time-summed response χ∗G=∑k=0∞AkB=(I−A)−1B. _ ^G= _k=0^∞A^kB=(I-A)^-1B. Under exact zero-order-hold discretization, A=eAzΔtA=e^A_z t and B=∫0ΔteAzsBzsB= _0 te^A_zsB_z\,ds, so (I−A)−1B=−Az−1Bz(I-A)^-1B=-A_z^-1B_z. This follows from ∫0ΔteAzss=Az−1(A−I) _0 te^A_zs\,ds=A_z^-1(A-I). An approximate propagator requires a consistent source discretization. The stable endpoint response is therefore δO=(DzO¯)χ∗δuorδO=(DzO¯)χ∗Gδu.δ O=(D_z O) _ δ u δ O=(D_z O) _ ^Gδ u. (S2.8) The graph checks in Section S7.1 test this composition under prescribed dynamics. They do not identify an autonomous language-model state. S2.3 Conditional well-posedness and coherence The following hypotheses give unique finite-horizon evolution from complete preparations, including internal histories and prescribed external inputs. Uniqueness does not imply attraction. For a finite maximum internal delay τmax _ , take the dynamical base space ℬdyn=H×∏iMiB_ dyn=H× _iM_i, with Banach memory spaces, and the phase space τ=C([−τmax,0],ℬdyn)X_τ=C([- _ ,0];B_ dyn) with the supremum norm. Write x(t)=(ct,m1(t),…,mN(t))x(t)=(c_t,m_1(t),…,m_N(t)) and ζt(θ)=x(t+θ) _t(θ)=x(t+θ). The prescribed continuous preparation ζ0 _0 has endpoint x(0)x(0). Internal quantities and their histories are specified functionals of this modeled history; any extra generator is included in ℬdynB_ dyn. Point evaluation is bounded in this phase space. Passive observation records are subsequently defined by their integrals when integrable, rather than assumed to belong to the dynamical phase space. The later coherence estimate additionally requires regularity and aggregate dissipativity. It is separate from finite-window response, readout boundaries and the branch assumptions in Section S2.2. Well-posed delayed field system. For any finite horizon [0,T][0,T]: 1. L generates a strongly continuous contraction semigroup on H; 2. the combined source and memory functionals are defined on all of τX_τ, are Lipschitz on each bounded history set and have at most linear growth in ‖ζ‖∞\|ζ\|_∞. Write their values as Ji(ζ)J_i(ζ) and Ψi(ζ,ηi) _i(ζ, _i), retaining all internal delay arguments. Global componentwise Lipschitz bounds suffice for finite compositions; local bounds require the separate growth condition; 3. the apparatus inputs ηi _i are strongly measurable and essentially bounded on finite horizons. Explicit time/input dependence of the functionals is strongly measurable at each fixed history, with Lipschitz and growth bounds uniform on finite horizons and bounded input sets; 4. each externally prescribed signal yℓy_ is supplied on [−τℓ,T][- _ ,T], is strongly measurable there and essentially bounded on that interval; and 5. each Eℓ:Yℓ→HE_ :Y_ → H is a bounded linear operator. Conditional mild well-posedness. Under the “Well-posed delayed field system” assumptions above, (S2.1) together with (S2.2) admits a unique mild solution on [0,T][0,T] for the compatible continuous preparation ζ0 _0, with external delayed signals treated as prescribed inputs. If the functionals are defined only on an open neighborhood, the conclusion holds only up to exit from that neighborhood unless continuation is separately established. Derivation. Variation of constants gives the field for 0≤t≤T0≤ t≤ T, with the negative-time history prescribed by ζ0 _0 40, 19, 18, 13: ct=etLc0+∫0te(t−s)L(∑iJi(ζs)+∑ℓEℓyℓ(s−τℓ))s.c_t=e^tLc_0+ _0^te^(t-s)L ( _iJ_i( _s)+ _ E_ y_ (s- _ ) )\,ds. The memory components satisfy the integral equations mi(t)=mi(0)+∫0tΨi(ζs,ηi(s))s.m_i(t)=m_i(0)+ _0^t _i( _s, _i(s))\,ds. Measurability and growth make the Bochner integrals well-defined. Contraction of the free semigroup and bounded-set Lipschitz continuity make their integral map contractive on a sufficiently small ball in C([−τmax,T];H×M1×⋯×MN)C([- _ ,T];H× M_1×·s× M_N), giving a unique local mild solution. For r(t)=sup−τmax≤s≤t‖x(s)‖r(t)= _- _ ≤ s≤ t\|x(s)\|, linear growth gives r(t)≤r(0)+at+b∫0tr(s)sr(t)≤ r(0)+at+b _0^tr(s)\,ds, hence r(t)≤(r(0)+at)ebtr(t)≤(r(0)+at)e^bt, with a,b≥0a,b≥ 0 on the fixed horizon. Global definition and bounded-set Lipschitz continuity therefore permit continuation to [0,T][0,T], without implying long-time boundedness or classical differentiability. This deterministic result covers the stated bounded prescribed inputs. White noise is not such an input: a white-noise-driven model requires a stochastic delay equation and a well-defined stochastic convolution under separate noise assumptions. An expected quadratic-energy estimate must include the Itô correction. For trajectories with the same additive noise coefficient, coupling them to the same noise realization cancels that term in their difference. The following functional measures field deviation, memory mismatch and recent external-input energy; a decay bound requires aggregate dissipativity. Coherence functional. Take Hilbert memory spaces MiM_i, and write c,mi,yℓc,m_i,y_ for deviations from an exact reference solution, including a fixed stationary operating point. A moving equilibrium adds a tracking residual as forcing and is excluded here. In the deviation equation, JiJ_i means the source difference between the two full trajectories, with their histories. Define C(t)=‖ct‖H2+∑i=1Nai‖mi(t)−Γict‖2+∑ℓ=1Nτbℓ∫t−τℓte−2α(t−s)‖yℓ(s)‖2s.C(t)=\|c_t\|_H^2+ _i=1^Na_i\|m_i(t)- _ic_t\|^2+ _ =1^N_τb_ _t- _ ^te^-2α(t-s)\,\|y_ (s)\|^2\,ds. (S2.9) Here ai,bℓ>0a_i,b_ >0, α>0α>0 is the discount rate, and Γi:H→Mi _i:H→ M_i is a bounded linear comparison map. The terms measure present field deviation, memory–field mismatch and discounted energy of the external yℓy_ histories. There is no norm of the full internal delay history ζt _t in C; in general, C does not control the norm in τX_τ. Bounded-coherence inequality. For sufficiently regular trajectories in a bounded neighborhood, let ei=mi−Γice_i=m_i- _ic and assume λ0>0 _0>0, dℓ≥0d_ ≥ 0 satisfy 2⟨c,Lc+∑iJi+∑ℓEℓyℓ(t−τℓ)⟩ 2 c,Lc+ _iJ_i+ _ E_ y_ (t- _ ) +2∑iai⟨ei,m˙i−Γic˙⟩ +2 _ia_i e_i, m_i- _i c ≤−λ0(‖c‖H2+∑iai‖ei‖2)+∑ℓdℓ‖yℓ(t−τℓ)‖2. ≤- _0 (\|c\|_H^2+ _ia_i\|e_i\|^2 )+ _ d_ \|y_ (t- _ )\|^2. This aggregate inequality is an additional hypothesis on the full source differences, including their internal delay dependence; neither well-posedness nor stability alone establishes it. If, in addition, bℓe−2ατℓ≥dℓb_ e^-2α _ ≥ d_ for every delay channel, then dtC(t)≤−λC(t)+∑ℓ=1Nτbℓ‖yℓ(t)‖2,λ=minλ0,2α>0. ddtC(t)≤-λ C(t)+ _ =1^N_τb_ \|y_ (t)\|^2, λ= \ _0,2α\>0. (S2.10) The estimate applies only while the stated regularity and aggregate dissipativity hypotheses hold. Derivation. Leibniz differentiation gives, for each history term, dt∫t−τℓte−2α(t−s)‖yℓ(s)‖2s=‖yℓ(t)‖2−e−2ατℓ‖yℓ(t−τℓ)‖2−2α∫t−τℓte−2α(t−s)‖yℓ(s)‖2s. ddt _t- _ ^te^-2α(t-s)\|y_ (s)\|^2\,ds=\|y_ (t)\|^2-e^-2α _ \|y_ (t- _ )\|^2-2α _t- _ ^te^-2α(t-s)\|y_ (s)\|^2\,ds. Adding these identities, weighted by bℓb_ , to the aggregate dissipativity bound and using bℓe−2ατℓ≥dℓb_ e^-2α _ ≥ d_ absorbs every delayed boundary term and yields (S2.10). The calculation uses the weighted-history differentiation underlying Lyapunov–Krasovskii estimates 18, 13; the aggregate dissipativity premise remains a separate assumption. Interpretation. The estimate bounds the selected deviations under its hypotheses; it is not a general delayed stability theorem. For example, the asymptotically stable scalar equation x˙(t)=−x(t)+0.4x(t−τ) x(t)=-x(t)+0.4x(t-τ), τ>0τ>0, with no auxiliary memories or external channels has C=x2C=x^2. A continuous initial history with x(0)=ε≠0x(0)= ≠ 0 and x(−τ)=3εx(-τ)=3 gives C˙(0+)=2ε(−ε+1.2ε)=0.4ε2>0. C(0^+)=2 (- +1.2 )=0.4 ^2>0. Stability permits this initial growth; the aggregate dissipation hypothesis fails for this preparation. A Lyapunov estimate for the full delayed state requires a functional controlling its internal histories and a separate dissipation bound. The estimate establishes neither controllability nor repair policy or cost. Empirical lag, deficit-area and recovery-fraction readouts summarize perturbation outcomes and do not test this differential inequality. S2.4 Readout-defined operating boundaries Criteria on scalar or vector readouts define regimes of the full state; under factorization they also define regimes on retained coordinates. For a nested pair, the branch and path assumptions below govern boundary encounters. Stationary susceptibility adds the closure and stability hypotheses of Section S2.2. S2.4.1 Local regimes and boundary crossings For nested workability and exactness readouts Pf,QfP_f,Q_f, the question is when a loading path has separated regime crossings. Here w=Pfw=P_f and r=Qf/Pfr=Q_f/P_f where Pf>0P_f>0; nesting alone does not order the crossings. Admissible loading path. Fix one admissible family f and let UadmU_adm be a local control manifold on which the regular operating branch u↦z∗(u)u z_ (u) exists smoothly. Call a C1C^1 path γ:[0,T]→Uadmγ:[0,T]→ U_adm an admissible loading path when it keeps the same instance class and the same outer/inner observables fixed, and when its readout sequence has no re-entry: after leaving r=1r=1 it never returns to r=1r=1, after reaching r=0r=0 it never returns to r>0r>0, and after reaching w=0w=0 it remains failed. These conditions define monotone loading but do not impose the relative locations of the first-hit thresholds. The path must follow the specified intervention and be fixed before crossing data are read; selection after observing the transition order is inadmissible. On the operating branch, these coordinates are w(u)=Pf(z∗(u))w(u)=P_f(z_ (u)) and r(u)=Qf(z∗(u))/Pf(z∗(u))r(u)=Q_f(z_ (u))/P_f(z_ (u)) for Pf(z∗(u))>0P_f(z_ (u))>0. The four regimes and their boundaries are defined together below. Local boundary surfaces. Define the four readout strata SE:=u∈Uadm:w(u)>0,r(u)=1,SPE:=u∈Uadm:w(u)>0, 0<r(u)<1,S_E:=\u∈ U_adm:\ w(u)>0,\ r(u)=1\, S_PE:=\u∈ U_adm:\ w(u)>0,\ 0<r(u)<1\, SW:=u∈Uadm:w(u)>0,r(u)=0,SF:=u∈Uadm:w(u)=0,S_W:=\u∈ U_adm:\ w(u)>0,\ r(u)=0\, S_F:=\u∈ U_adm:\ w(u)=0\, the exact, partially exact, workable-but-inexact, and failed strata. An overline on a set denotes its closure, including its limit points. Define each transition locus as the common relative boundary of the two adjacent strata, ΣEW:=SE¯∩SPE¯,ΣWR:=SPE¯∩SW¯,ΣRO:=SE∪SPE∪SW¯∩SF¯. _EW:= S_E∩ S_PE, _WR:= S_PE∩ S_W, _RO:= S_E∪ S_PE∪ S_W∩ S_F. The closures for ΣEW _EW and ΣWR _WR are taken relative to U+=w>0U_+=\w>0\, and the closure for ΣRO _RO relative to UadmU_adm. Assume each locus is locally a C1C^1 codimension-one hypersurface in this relative topology and the loading path meets it transversally at nonzero rate. At each encounter, additionally require the path to pass between the two defining strata on nonempty adjoining intervals. This is an actual change of regime, not merely an intersection with their closures; for ΣRO _RO, the two sides are workable and failed. These assumptions can fail at plateau edges, intersections or corners. The set identities alone establish neither smoothness, actual regime changes nor encounter order. The parameters sEWs_EW, sWRs_WR and sROs_RO are the first encounters with the corresponding transition loci along the declared path; an unencountered locus has first-hit parameter +∞+∞. The four-step statement requires all three encounters within the observation interval. Along a continuous admissible monotone path starting in the exact stratum, the following order holds only when the first-hit parameters satisfy sEW<sWR<sROs_EW<s_WR<s_RO: exact stratum→partially exact region→workable-but-inexact stratum→failed stratum.exact stratum exact region -but-inexact stratum stratum. Nesting gives 0≤r≤10≤ r≤ 1 on U+U_+; failure is specified by w=0w=0, where r is undefined. For discontinuous readouts, the declared equality convention assigns threshold points. Smooth boundaries and transverse encounters that change the adjacent strata are additional assumptions. The no-re-entry condition, actual crossings and ordered finite encounters sEW<sWR<sROs_EW<s_WR<s_RO give the displayed sequence, with nonempty intervals in each adjacent regime. Coincident thresholds can skip a regime; they are non-generic only if the threshold conditions vary independently. Regularity and transversality isolate encounters but do not order them. Thus the sequence is specific to the declared family and path (main-text Figure 3). Boundary-test selection principle. For a fixed instance class, select controls whose projected responses distinguish exactness loss from workability loss while preserving the stated no-re-entry conditions along the same loading path. Thresholds, response curves and instance locations require empirical calibration. S2.5 Susceptibility and comparison of macroscopic readouts The question is when a finite-window field response admits a space–time kernel or stationary limit and how it projects to an observable. The conditions below govern these steps; h denotes the perturbing source history. Susceptibility operator and kernel. For a declared source space U and field space ℋH, the primary response is the bounded derivative ℛt=Dh⟨c(t)⟩:→ℋR_t=D_h c(t) :U , when this Fréchet derivative exists. Angle brackets denote ensemble averages; α,βα,β index field channels. If this operator admits a space–time kernel, its channel components are written χαβ(,,t,t′)=δ⟨cα(,t)⟩δhβ(,t′). _αβ(x,y;t,t )= δ c_α(x,t) δ h_β(y,t ). This notation denotes a weak or distributional kernel unless additional spatial regularity gives pointwise values. In particular, point evaluation is not a bounded readout on L2L^2, and a bounded operator on that space need not have an ordinary integral kernel. For c˙=−c+h c=-c+h, the impulse response is e−(t−t′)Ie^-(t-t )I for t≥t′t≥ t , and zero earlier; its spatial kernel is a delta distribution. Bounded linear readouts act directly on ℛtR_t; an L2L^2 readout weight gives the corresponding inner product. Pointwise readouts require a stronger state space or a separately justified regularization. For a deterministic finite graph G, or a pathwise derivative with the noise realization held fixed, define the field and readout responses by χ^αβG(x,y,t,t′)=∂cα(x,t)∂hβ(y,t′),χ^ajred=∂Oa∂θj. χ^G_αβ(x,y;t,t )= ∂ c_α(x,t)∂ h_β(y,t ), χ red_aj= ∂ O_a∂ _j. Here x,y∈VGx,y∈ V_G are vertices of G=(VG,EG)G=(V_G,E_G), t,t′t,t are discrete times, and α,βα,β index the declared finite channel basis; cα(x,t)c_α(x,t) is the field-channel value. The amplitude hβ(y,t′)h_β(y,t ) multiplies a unit source direction at the specified vertex, channel and time. OaO_a is a smooth readout component, and θj _j is the amplitude along a specified source-control direction. When the readout depends only on the field coordinates represented in this operator, applying the source directions and readout differential gives χ^red χ red. Readouts that also use memory or trajectory records require the corresponding augmented-state response. This finite estimate becomes a continuum kernel only under graph, time and channel refinement, and a stationary susceptibility only under the stable limit below. For mean responses, differentiate the ensemble mean; averaging pathwise derivatives is valid only when derivative and expectation interchange. Solution, projection and readout differentiability are additional to Lipschitz well-posedness. Nonsmooth points require finite contrasts or specified one-sided responses. For fixed preparation, including field, memories and histories, let sTs_T be the full terminal state and zT=Gf(sT)z_T=G_f(s_T) its retained description. If the readout has a differentiable representation OT(sT)=O¯T(zT)O_T(s_T)= O_T(z_T), the chain rule gives DsOT(sT)=DzO¯T(zT)DsGf(sT).D_sO_T(s_T)=D_z O_T(z_T)\,D_sG_f(s_T). Here DD denotes the derivative: a Jacobian in finite dimensions or a Fréchet derivative on the specified function space. The bar labels the readout as a function of zTz_T, not an ensemble average. Read from right to left, the derivatives map a full-state perturbation to a retained-state change and then to a readout change. If the graph contains the full required augmented state and is itself retained, GfG_f is the identity and DsGf=ID_sG_f=I; otherwise keep this derivative in the response chain (Section S1.3). For a graph used as the retained state or a verified closed retained transition, let A,BA,B be the time-independent retained transition and source maps. If ρ(A)<1ρ(A)<1, a persistent tangent source gives the resolvent χ∗G _ ^G derived in Section S2.2. A one-time source at t′t instead gives the retarded Green function At−t′BA^t-t B for t≥t′t≥ t and zero response earlier. The discrete resolvent acts on retained coordinates; a full-state response must first be mapped to retained coordinates by DsGfD_sG_f. Equality with the continuous-time branch susceptibility in Eq. (S2.6) requires exact zero-order-hold discretization (Section S2.2) or a demonstrated graph-and-time refinement limit. Section S7.1 gives response checks for code-repair constructions using endpoint information, deterministic replay and readouts from fresh model calls. The participation ratio rPRr_ PR defined in Section S1.6 measures concentration in the response matrix’s squared singular values. It equals matrix rank when the nonzero singular values are equal, and approaches one when a single squared singular value dominates their sum. This spectral diagnostic does not determine the minimal state needed for an endpoint readout or a common dimensionality across realizations. The continuum comparison below preserves source units and controls accumulated response. Let hn=Δtn→0h_n= t_n→ 0, ρ(An)<1ρ(A_n)<1, and let BnB_n be a per-unit-time source operator, not an integrated source step. Interpolation InI_n maps graph to continuum states; PnUP_n^U maps the declared source space to graph inputs. The complete linearized continuum generator ℒL includes source, memory and feedback derivatives; it differs from the medium generator L in Eq. (S2.1) unless those contributions vanish. Assume it generates a strongly continuous semigroup, 00 is in its resolvent set and the corresponding continuum source operator ℬB, distinct from the graph source map B, is bounded. On the same declared spaces, sufficient joint consistency and stability conditions, in source-space operator norm, are convergence for each t≥0t≥ 0 and a bound uniform in all n,kn,k: InAn⌊t/hn⌋BnPnU→etℒℬ,∥InAnkBnPnU∥≤Ce−ωkhn,C,ω>0.I_nA_n t/h_n B_nP_n^U→ e^tLB, \|I_nA_n^kB_nP_n^U\|≤ Ce^-ω kh_n, C,ω>0. The identity hn∑k≥0InAnkBnPnU=∫0∞InAn⌊t/hn⌋BnPnUth_n _k≥ 0I_nA_n^kB_nP_n^U= _0^∞I_nA_n t/h_n B_nP_n^U\,dt and dominated convergence then give In(I−An)−1hnBnPnU→(−ℒ)−1ℬ.I_n(I-A_n)^-1h_nB_nP_n^U→(-L)^-1B. Joint consistency controls sources propagated through hidden coordinates; separate projected consistency of An,BnA_n,B_n is insufficient. These are refinement hypotheses, not a continuum-limit conclusion for a reported graph. Matched susceptibility and readout response. For matched perturbation channels and the same fixed linear readout, equal mean-field susceptibilities give equal first-order mean endpoint responses. The same statement holds for smooth readouts linearized about a deterministic reference with the same readout differential. Equality of the susceptibility alone does not imply agreement under different readout maps. Derivation. Let Kα(,t)K_α(x,t) be the fixed readout weighting kernel, with summation over repeated channel indices. For the linear coarse observable O[c](t)=∫Kα(,t)cα(,t)O[c](t)= K_α(x,t)c_α(x,t)\,dx, δ⟨O(t)⟩=∫Kα(,t)χαβ(,,t,t′)δhβ(,t′)dt′.δ O(t) = K_α(x,t) _αβ(x,y;t,t )δ h_β(y,t )\,dx\,dy\,dt . The conclusion follows by contraction with the common readout kernel. A smooth readout linearized about a deterministic reference has the same first-order form with K=DO|c∗K=DO|_c_ . For nonlinear observables of a fluctuating field, the ensemble response requires a separately justified closure or the joint state and response statistics. The statement is conditional on matched channels, coarse-graining and readout semantics; none of the empirical realizations is claimed to share a measured susceptibility, response spectrum, dimension or boundary geometry. S3 Physical measurement methods and additional results For the models and observables in main-text Section 3, the methods below compare simulated delayed consensus with analytic response predictions, then examine distinct measures of flock order. They specify estimators, uncertainty and the interpretation of each comparison. S3.1 Driven delayed consensus network Simulated network trajectories, one per delay and seed, provide the collective mean signal and spatial variance. The mean’s amplitude and phase are compared with the harmonic response of the delayed mean equation; separate drive/noise sweeps test the workability and exactness criteria. Numerical trajectory and readout. The delayed network and harmonic target are defined in main-text Eqs. (10)–(14). The independent unit-intensity white noises satisfy ⟨ξi(t)⟩=0 _i(t) =0 and ⟨ξi(t)ξj(t′)⟩=δijδ(t−t′) _i(t) _j(t ) = _ijδ(t-t ), with Kronecker and Dirac deltas, respectively; angle brackets denote ensemble averages. The documented Euler–Maruyama scheme evaluates the drift at the beginning of each step and adds independent Gaussian increments of standard deviation σΔtσ t at each site. Linear interpolation of stored states approximates c(t−τ)c(t-τ). The documented initialization is ci(t)=0.01gic_i(t)=0.01g_i for −τ≤t≤0-τ≤ t≤ 0, with independent standard-normal gig_i. A Mersenne Twister generator is reset for each trajectory; its stream supplies the initial values and subsequent noise. The calibration tests local harmonic-response agreement in a stable weak-delay regime. The 1313-delay sweep spans τ∈[0.43,0.46]τ∈[0.43,0.46], using window [300,400][300,400], Ω=0.35 =0.35 and Δt=0.02 t=0.02. The window length Twin=100T_ win=100 gives the estimator (2Δt/Twin)∑n=1500020000m(nΔt)e−iΩnΔt(2 t/T_ win) _n=15000^20000m(n t)e^-i n t, with 5,0015,001 samples. Amplitude and phase lag are averaged over seeds 20260432012026043201–20260433002026043300, reused at each delay, then compared with the harmonic target. The records specify linear interpolation, whereas their cited simulator revision rounds delay to a grid step. A matching executed-source record and runtime environment are absent: the stated scheme is a documented reconstruction with incomplete execution provenance. Coverage is limited to Ωτ∈[0.1505,0.1610] τ∈[0.1505,0.1610], with a>|K|a>|K| throughout. Workability remains saturated; the drive/noise sweeps below test nested criteria. The calibration includes neither matched preparations with equal current fields and different histories nor comparative history-compression tests. Observation accumulators give terminal readouts of an enlarged state (Section S1.2); they do not establish autonomous three-dimensional dynamics. The calculation retains delay history and assumes neither a slow manifold nor a time-summed resolvent. Uniform forcing and mean readout select the consensus Fourier mode. Circulant unit-row-sum coupling preserves that mode and its zero-sum complement, whose N−1N-1 transverse modes have zero uniform-source response and mean-readout sensitivity to numerical precision. The deterministic spatial response is therefore rank one, while its evolution still requires delay history. The following noise calculation explains what the additional variance readout measures. Projection of the noise. Normal modes separate collective noise in the lock-in estimator from transverse noise measured by V. The variance scale (σ2|χ|2/(NTwin))O(σ^2|χ|^2/(NT_ win)) below concerns complex lock-in fluctuations, not amplitude or phase error. (i) Normal-mode decomposition. Write ci(t)=m(t)+δci(t)c_i(t)=m(t)+δ c_i(t) with ∑iδci=0 _iδ c_i=0. Here δciδ c_i denotes the spatial fluctuation about the instantaneous mean, rather than a source-response perturbation. The symmetric ring averaging matrix preserves the consensus and zero-sum transverse subspaces. Main-text Eq. (11) gives the exact mean projection, including its noise ηm _m. (i) Collective-noise intensity σ2/Nσ^2/N. Independence of the ξi _i gives ⟨ηm(t)⟩=0,⟨ηm(t)ηm(t′)⟩=σ2N2∑i,j⟨ξi(t)ξj(t′)⟩=σ2Nδ(t−t′), _m(t) =0, _m(t)\, _m(t ) = σ^2N^2 _i,j _i(t) _j(t ) = σ^2N\,δ(t-t ), (S3.1) Thus spatial averaging reduces the noise amplitude from σ to σ/Nσ/ N. (i) Lock-in as a finite-window matched-filter estimator. Main-text Eq. (11) is linear, so m=mdet+mnoisem=m_ det+m_ noise. After transients, the periodic component is mdet(t)=ReFχeiΩtm_ det(t)=Re\Fχ e^i t\, with χ=χ(Ω,τ)χ=χ( ,τ); χ¯ χ denotes its complex conjugate. On a rectangular window [t0,t0+Twin][t_0,t_0+T_ win], its lock-in is Z^det=Fχ+Fχ¯e−2iΩ(t0+Twin/2)sin(ΩTwin)ΩTwin. Z_ det=Fχ+F χ\,e^-2i (t_0+T_ win/2) ( T_ win) T_ win. The second term is the contribution from the conjugate frequency of the real signal. It vanishes when ΩTwin∈πℤ T_ win∈ , and decays as the window grows. Thus FχFχ, with amplitude th=F|χ|A_ th=F|χ| and phase lag ϕth=|argχ| _ th=| χ|, is the harmonic-response target rather than the exact value of every finite window estimator. The recorded experiment uses the discrete rectangular sum Z^grid(τ)=2ΔtTwin∑j=05000ReFχ(Ω,τ)eiΩtje−iΩtj,tj=300+jΔt,Δt=0.02. Z_ grid(τ)= 2 tT_ win _j=0^5000Re\Fχ( ,τ)e^i t_j\e^-i t_j, t_j=300+j t, t=0.02. Set grid=|Z^grid|A_ grid=| Z_ grid| and ϕgrid=|argZ^grid| _ grid=| Z_ grid|. The finite window accounts for most of the phase discrepancy across the 13 delays: ϕgrid−ϕth=0.674400 _ grid- _ th=0.674400–0.676273∘0.676273 , whereas the recorded seed-mean phase minus ϕgrid _ grid is 0.0586750.058675–0.058919∘0.058919 . These signed terms add. The steady periodic solution of the documented Euler–Maruyama scheme adds 0.0541150.054115–0.054358∘0.054358 to ϕgrid _ grid on the same window; the residual seed-mean difference is approximately 0.00456∘0.00456 . This analytic discretization correction assumes that scheme; it does not test time-step convergence. The amplitude corrections partially cancel: the grid-minus-harmonic and seed-mean-minus-grid differences, each divided by thA_ th, are −0.003809-0.003809 to −0.003710-0.003710 and +0.001292+0.001292 to +0.001299+0.001299. These are analytic reanalyses of the stated estimator and stored means. The amplitude remainder combines discretization, residual transients and stochastic averaging, whose contributions are unresolved. Stochastic uncertainty is a separate contribution. The stochastic part mnoisem_ noise is the output of the stable filter H(ω)=(a−Ke−iωτ+iω)−1H(ω)=(a-Ke^-iωτ+iω)^-1 (the continuation of χ to general ω) driven by white noise of intensity σ2/Nσ^2/N; by Wiener–Khinchin its power spectral density is the smooth bounded Sm(ω)=(σ2/N)|H(ω)|2S_m(ω)=(σ^2/N)|H(ω)|^2. The lock-in integral is a Fourier estimator with bandwidth ∼1/Twin 1/T_win. For stationary noise with integrable autocovariance Rm(v)=(2π)−1∫Sm(ω)eiωvωR_m(v)=(2π)^-1 S_m(ω)e^iω v\,dω, the complex variance means VarℂZ=|Z−Z|2Var_CZ=E|Z-EZ|^2. The rectangular-window integral has variance 4Twin−2∫−TwinTwin(Twin−|v|)Rm(v)e−iΩvv4T_win^-2 _-T_win^T_win(T_win-|v|)R_m(v)e^-i v\,dv, hence Varℂ(Z^noise)=4σ2|χ(Ω,τ)|2NTwin+o(Twin−1).Var_C( Z_ noise)= 4σ^2|χ( ,τ)|^2NT_win+o(T_win^-1). (S3.2) The remainder o(Twin−1)o(T_ win^-1) decays faster than 1/Twin1/T_ win. This asymptotic expression concerns the continuous rectangular-window estimator; the recorded discrete sum has its corresponding discrete covariance sum. Spatial and temporal averaging suppress stochastic variance. The total discrepancies reported in main-text Table 3 also include demodulation leakage, transients and discretization. (iv) Transverse modes as the retained variance V. Subtracting main-text Eq. (11) from main-text Eq. (10) removes the uniform drive and consensus-mode dynamics exactly, leaving the transverse fluctuations forced by the transverse projection of the noise, δc˙i(t)=−aδci(t)+K|Ni|∑j∈Niδcj(t−τ)+σδξi(t),δξi≡ξi−1N∑kξk. δ c_i(t)=-a\,δ c_i(t)+ K|N_i| _j∈ N_iδ c_j(t-τ)+σ\,δ _i(t), δ _i≡ _i- 1N _k _k. (S3.3) On the transverse coupling eigenmodes eqq≠0\e_q\_q≠ 0 (eigenvalues κq∈[−1,1) _q∈[-1,1); the consensus mode κ0=1 _0=1 is excluded), each amplitude δcqδ c_q is a stable scalar delay Ornstein–Uhlenbeck process whose stationary variance is the spectral integral ⟨δcq2⟩=σ22π∫−∞|a−Kκqe−iωτ+iω|−2ω=:σ2gq(τ),gq(τ)>0, δ c_q^2 = σ^22π _-∞^∞ |a-K _qe^-iωτ+iω |^-2\,dω=:σ^2\,g_q(τ), g_q(τ)>0, (S3.4) where gq(0)=1/(2aeff,q)g_q(0)=1/(2a_ eff,q) is the memoryless Ornstein–Uhlenbeck factor and aeff,q=a−Kκq>0a_ eff,q=a-K _q>0 for the stated parameters. By Parseval, the stationary expectation of the spatial variance is ⟨V(t)⟩=1N∑i=1N⟨(ci−m)2⟩=1N∑q≠0⟨δcq2⟩=σ2N∑q≠0gq(τ)=(σ2). V(t) = 1N _i=1^N (c_i-m)^2 = 1N _q≠ 0 δ c_q^2 = σ^2N _q≠ 0g_q(τ)=O(σ^2). (S3.5) The reported V estimates this stationary expectation by averaging the instantaneous spatial variance over the finite observation window. Thus transverse noise gives ⟨V(t)⟩=(σ2) V(t) =O(σ^2), while collective noise contributes at order σ2|χ|2/(NTwin)σ^2|χ|^2/(NT_ win) to complex lock-in variance (Eq. (S3.2)), with subleading amplitude/phase effects. The analytic amplitude is th=F|χ(Ω,τ)|A_ th=F|χ( ,τ)|. The stationary variance relative to its square is ⟨V(t)⟩th2=g¯(τ)σ2F2|χ(Ω,τ)|2,g¯(τ)=1N∑q≠0gq(τ)=(1), V(t) A_ th^2= g(τ)\, σ^2F^2|χ( ,τ)|^2, g(τ)= 1N _q≠ 0g_q(τ)=O(1), (S3.6) the mode-averaged factor of Eq. (S3.5). In the stated simulation units, the noise-to-drive amplitude ratio σ/F=0.06σ/F=0.06 sets the prefactor σ2/F2≈3.6×10−3σ^2/F^2≈ 3.6× 10^-3, and the measured value across the delay window is V/2<10−3V/A^2<10^-3, i.e. 2/(2+V)≥0.999A^2/(A^2+V)≥ 0.999. Meaning of the spatial summary. The concentration ratio above uses peak amplitude squared rather than its time average. A sinusoid has mean square 2/2A^2/2, so this is not a time-averaged energy fraction. A participation ratio follows from nonnegative weights E0=2,∑k=1N−1Ek=V,reff=(∑k≥0Ek)2∑k≥0Ek2.E_0=A^2, _k=1^N-1E_k=V, r_ eff= ( _k≥ 0E_k)^2 _k≥ 0E_k^2. Writing C=2/(2+V)C=A^2/(A^2+V), concentration of all transverse weight in one mode and equal distribution over N−1N-1 modes give 1C2+(1−C)2≤reff≤1C2+(1−C)2/(N−1). 1C^2+(1-C)^2≤ r_ eff≤ 1C^2+(1-C)^2/(N-1). At the least-concentrated delay, C≈0.9990C≈ 0.9990 and N=64N=64. Using the unrounded recorded concentration gives the outward-rounded bounds 1.00199≤reff≤1.002001.00199≤ r_ eff≤ 1.00200, summarized as reff≈1.002r_ eff≈ 1.002. These bounds are derived from recorded amplitude and variance, not measured covariance spectra at all 13 delays. Their lower endpoint is not a lower bound for every delay. Let NsampN_ samp be the number of observation times. To express these same weights through eigenvalues μk _k of the sampled second-moment matrix Nsamp−1∑jδc(tj)δc(tj)⊤N_ samp^-1 _jδ c(t_j)δ c(t_j) , use Ek=αμk/NE_k=α _k/N, where α=NsampΔt/Twin=5001/5000α=N_ samp t/T_ win=5001/5000. This factor matches the rectangular-quadrature definition of V; for seed-averaged records, average these matrices over the same seeds. Unscaled second-moment weights have a different normalization. The uniform drive excites only the consensus spatial mode. Its dominance is a spatial-response summary, not the minimal dimension of an autonomous three-dimensional state. Drive and noise sweeps of the operating criteria. The sweeps compare an admissible signal with the stricter amplitude-and-phase criterion as drive decreases or noise increases. At τ=0.43τ=0.43, main-text Eq. (15) defines PfP_f by finite-valued integration through the horizon, a nonempty measurement window, amplitude ≥0.16A≥ 0.16 and variance V≤0.05V≤ 0.05; QfQ_f additionally requires ≥0.22A≥ 0.22 and phase lag ϕ≤35∘φ≤ 35 . Decreasing drive reduces the coherent signal at fixed noise σ=0.015σ=0.015; increasing noise raises transverse variance and perturbs phase at fixed drive F=0.25F=0.25. Both retrospective sweeps reuse seeds 20260428012026042801–20260428502026042850 at every control value with fixed thresholds; their records bind the simulator source for the stated update and initialization. Figure S1: Sample support fractions for nested consensus criteria. Fractions P^f P_f (circles) and Q^f Q_f (squares) of the same 50 simulation seeds meeting the outer and stricter nested criteria, respectively, at all 13 recorded points on each axis. a, Drive decreases at fixed noise σ=0.015σ=0.015. b, Noise increases at fixed drive F=0.25F=0.25. Both use delay τ=0.43τ=0.43. Shading between curves gives the fraction passing only the outer criterion at each sampled point. Lines interpolate points for display; the dotted line marks half support. The outer rule requires finite-valued integration through the horizon, a nonempty measurement window, lock-in amplitude at least 0.16 and window-averaged spatial variance at most 0.05; the inner rule also requires amplitude at least 0.22 and phase lag at most 35∘35 . Nesting imposes Qf≤PfQ_f≤ P_f; the measured fractions quantify the sample gap. Reused noise realizations correlate the points. In panel b the early half-support crossing is consistent with finite-sample phase fluctuations; it does not identify a population phase-loss boundary. Axes use the simulation’s field and time units. Complete fractions, amplitude and variance means remain in the authors’ records. To quantify criterion separation, average the binary readouts over 5050 seeds to obtain support fractions P^f P_f and Q^f Q_f. Linear interpolation gives half-support point estimates (fraction equal to one half): under decreasing drive, F≈0.163F≈ 0.163 for QfQ_f and F≈0.118F≈ 0.118 for PfP_f; under increasing noise, the corresponding estimates are σ≈0.138σ≈ 0.138 and 0.3100.310. Figure S1 shows both fractions on the complete recorded grids. Across 2626 control values and 5050 seeds each, 442442 of the 1,3001,300 trajectories are workable but inexact, Qf=0<PfQ_f=0<P_f, with no Qf>PfQ_f>P_f violations. Nesting guarantees weak order on both paths; measurements quantify interpolated sample separation and workable-but-inexact frequency. Drive crossings mainly reflect amplitude thresholds. On the noise axis, phase failure accounts for the workable-but-inexact trajectories: 179 of 650 satisfy Pf=1,Qf=0P_f=1,Q_f=0, all passing the extra amplitude criterion. All outer failures exceed the variance threshold: another 150 have Pf=0P_f=0, forcing Qf=0Q_f=0; 68 of these have phase at or below 35∘35 . Phase failure alone therefore cannot explain all 329 cases with Qf=0Q_f=0. Across 20,00020,000 paired-seed bootstrap resamples, central 95% ranges for the interpolated QfQ_f half-support statistic are [0.1625,0.1629][0.1625,0.1629] on the drive axis and [0.075,0.303][0.075,0.303] on the noise axis. These describe variation for this grid, window and empirical seed distribution, without established population-threshold coverage. The resampled crossing can reflect variance failures even when phase support exceeds one half; the statistic identifies neither a separate population phase boundary nor an outer threshold between controls whose outcomes all change together. Sampling dependence and population phase probability. Reusing the same 50 random-number streams makes the noise controls strongly dependent. At σ=0.10,0.15,0.20,0.25,0.30σ=0.10,0.15,0.20,0.25,0.30, respectively, the phase-pass counts are 28,24,24,23,2328,24,24,23,23; all 50 runs pass the outer criterion and the extra amplitude criterion. The corresponding pointwise 95% Wilson intervals at 0.100.10, 0.150.15 and 0.300.30 are [0.423,0.688][0.423,0.688], [0.348,0.615][0.348,0.615] and [0.330,0.596][0.330,0.596]. They are not simultaneous intervals. Across all pairs of noise controls, the Pearson correlations of the 50 phase values exceed 0.999690.99969; the phase-pass vectors at 0.150.15 and 0.200.20 are identical. Repeated sub-half fractions therefore do not supply independent replications. Linearity explains this dependence and permits a population calculation. After negligible initialization transients, write the numerical lock-in as Z^=Z0+σζ Z=Z_0+σζ, where Z0Z_0 is the deterministic numerical response and ζ is a zero-mean complex Gaussian variable whose two real components have covariance fixed by the linear evolution and window. A fixed seed reuses the same ζ at every noise amplitude. For the stated Euler step and linearly interpolated delay at τ=0.43τ=0.43, the mean obeys mn+1=0.98mn+0.004(mn−21+mn−22)+0.005cos(0.007n)+σ0.02/64ϵn,m_n+1=0.98m_n+0.004(m_n-21+m_n-22)+0.005 (0.007n)+σ 0.02/64\, _n, where ϵn _n are independent standard normal variables. Applying the recorded lock-in sum gives Z0≈0.280127897−0.195222004iZ_0≈ 0.280127897-0.195222004i, with phase lag 34.872761∘34.872761 . Applying that sum to the continuous harmonic solution instead gives 34.818645∘34.818645 ; these are different deterministic references. Let θ=35∘θ=35 , d=ReZ0sinθ+ImZ0cosθd=ReZ_0 θ+ImZ_0 θ, and let s2s^2 be the variance of Reζsinθ+ImζcosθReζ θ+Imζ θ. The lower edge of the allowed phase sector is a linear Gaussian inequality. The opposite edge has negligible tail probability over the recorded range, so, with ΦN _ N the standard-normal cumulative distribution, Prϕ≤35∘≃ΦN(dσs),d≈0.000758257,s≈0.0241341. \φ≤ 35 \ _ N\! ( dσ s ), d≈ 0.000758257, s≈ 0.0241341. The covariance follows from the impulse response of the recurrence above, without fitting the seed outcomes. This probability is approximately 0.5830.583 at σ=0.15σ=0.15 and 0.5420.542 at 0.300.30, remaining above one half throughout the sampled range. The observed early crossing at 0.13750.1375 is consistent with finite-sample fluctuations about this population response. This conclusion concerns the phase condition: the full probabilities Pr(Qf=1) (Q_f=1) and Pr(Pf=1) (P_f=1) also include the variance constraint. Their half-support points can remain ordered near the variance-driven decline; the records do not establish the absence of any population QfQ_f-before-PfP_f crossing. S3.2 Field-driven Vicsek flock Flock trajectories compare polarization PfP_f, alignment among particles, with QfQ_f, the fraction within a specified angle of the imposed field. Noise sweeps test whether these nonnested readouts decrease on the same scale. Standard continuum descriptions retain density and vector polarization 53, 31, 43, 8. Sampling and angular-noise convention. Main-text Eqs. (16)–(18) specify the update, parameters and order of readout averaging. The independent uniform angular noises have variance Var(ξi)=(2π)2/12=π2/3Var( _i)=(2π)^2/12=π^2/3. Seeds 2121–2424 use 500500 burn-in and 300300 measurement steps. The documented initialization draws positions uniformly in the square and headings uniformly on [−π,π)[-π,π). No observation-time or fixed-density size limit is inferred. The loading parameter b=ηb=η varies at fixed field, and the alignment tolerance is compared over 20∘,30∘,45∘\20 ,30 ,45 \. On η∈[0.15,0.70]η∈[0.15,0.70], measured Qf≤PfQ_f≤ P_f; r=Qf/Pfr=Q_f/P_f for Pf>0P_f>0 compares alignment with polarization, not conditional exactness. Order fails outside this interval: near disorder QfQ_f approaches random-orientation baseline 1/61/6 while polarization has a finite-size residual; at low nonzero noise QfQ_f can also exceed PfP_f. These nonnested readouts are excluded from workability–exactness regimes. The wrapped-normal comparison below is a descriptive approximation to the steady-state readouts, not a measured dynamical branch. Separated sampled readout scales. Across η∈[0,0.85]η∈[0,0.85], polarization decreases 1.000→0.0751.000→ 0.075 and alignment 1.000→0.1781.000→ 0.178. For seeds 2121–2424, first sampled half-values occur at η=0.35η=0.35 (alignment) and 0.600.60 (polarization) (main-text Fig. 4e). On [0.15,0.70][0.15,0.70], Qf≤PfQ_f≤ P_f but r is nonmonotone, excluding a global loading law or four-regime trajectory. At tolerances 20∘,30∘,45∘20 ,30 ,45 , all four seeds give alignment half-values 0.25,0.35,0.500.25,0.35,0.50, each below 0.600.60. The 45∘45 gap is 0.100.10, two grid steps. The wrapped-normal summary places coincident half-values at 45.50∘45.50 , close to reversal: separation is specific to the tested thresholds. For η=0.35η=0.35 to 0.550.55, alignment within 30∘30 is below one half while polarization stays above it. Wrapped-normal comparison. A wrapped-normal distribution gives a simple approximate relation between polarization and the fraction aligned with the field. Let σθ _θ be the standard deviation of the Gaussian angle before wrapping. For a wrapped-normal heading distribution centered on the imposed field, the population resultant is R=|eiθ|=e−σθ2/2R=|Ee^iθ|=e^- _θ^2/2. The measured PfP_f instead averages the instantaneous finite-flock modulus; it is not generally R. Setting R=PfR=P_f defines an effective summary, with inferred width (for 0<Pf<10<P_f<1) σθ(P)(η)=−2logPf(η). _θ^(P)(η)= -2 P_f(η). Let ΦN _N be the standard-normal cumulative distribution function. The corresponding probability for the principal wrapped angle to lie within the tolerance is QfWN(η)=∑k∈ℤ[ΦN(tol+2πkσθ(P)(η))−ΦN(−tol+2πkσθ(P)(η))],tol=30∘.Q_f WN(η)= _k [ _N\! ( tol+2π k _θ^(P)(η) )- _N\! ( -tol+2π k _θ^(P)(η) ) ], =30 . Using the measured PfP_f values to infer σθ(P) _θ^(P), with no fit parameter, gives RMSE(Qf,QfWN)=0.0507RMSE(Q_f,Q_f WN)=0.0507 across the 12 equally weighted noise-point means on η∈[0.15,0.70]η∈[0.15,0.70]. It overpredicts alignment at 11 of 12 points, with mean signed difference 0.04430.0443 for QfWN−QfQ_f WN-Q_f and maximum absolute difference 0.07170.0717. Finite-size effects, collective-direction fluctuations and distributional mismatch can contribute to these residuals; their separate contributions are unresolved. A perfectly coherent flock can turn away from the field while its polarization stays one. This is a descriptive comparison of the two curves, not a distributional fit test, a measured dynamical state or a fluctuation–relaxation law. Comparing normalized response profiles. An exploratory comparison tests whether flock and replayed AI response curves agree beyond a shared normalization. It compares the flock’s field-alignment fraction with the correctly decoded field fraction from recorded AI events, varying acceptance thresholds at fixed replayed graph scores. For each profile Q(b)Q(b), where b is noise or acceptance threshold, set S(b)=Q(b)/Q(0)S(b)=Q(b)/Q(0), with Q(0)>0Q(0)>0. Let b1/2b_1/2 be the first interpolated half-value and d<0d<0 the secant slope of S through its bracketing samples. The dimensionless coordinate λ=|d|(b1/2−b)λ=|d|(b_1/2-b) fixes an anchor and slope along S=12+λS= 12+λ by construction; common dynamics and a shared two-sided tangent require separate evidence. Residuals also include sampling and interpolation effects. Within windows |λ|≤W|λ|≤ W of half-width W=0.25W=0.25 or less, the sole recorded AI loading level leaves its reference-line residual undefined because reporting requires at least two samples in the window. At half-width 0.10, the two flock samples define the secant, imposing zero residual. The control uses only recorded profiles, with no new model calls or independent physical realizations, and supplies no evidence of a universal response shape. S4 Methods for inverse tests under fixed inputs The inverse tests ask whether a candidate state preserves later responses and, if rejected, which history distinctions it must retain. A known deterministic program permits exact execution checks; a sampled model requires output-law comparisons with uncertainty bounds. The methods for main-text Section 5 therefore distinguish exact-program reconstruction from bounded support on measured histories. In both tests, two sensors S1,S2S_1,S_2 are assigned one-to-one to recipients A,BA,B. The two association orientations exchange these assignments. Time zero is the observation cut: requests precede it and returns follow it. Pairs of request times or returned values are ordered by (S1,S2)(S_1,S_2), independently of record order. S4.1 Exact request–return matching The constructed task tests which pending-request information predicts a recipient’s later quantity. Histories associate sensors with recipients; continuations supply values, arrival times and a query. The known program matches returns to requests and reports a quantity and threshold indicator. Reconstruction compares matched candidate states under common continuations against the exact readout rule (main-text Sections 5.1–5.2). The selected representation and hashes of the protocol, implementation and discovery records are saved before a separate invocation for new comparisons. These bind local records, without independent registration or external chronology. Discovery uses four histories combining two association orientations and two notes, issue times (−4,−2)(-4,-2), query A, values (2,7)(2,7), and S1S_1 returning first. Each snapshot candidate has four violating pairs among six; the association state has two matched pairs without violations, and full history has none. Minimizing state count after excluding violations selects the association description, with ties resolved by fixed candidate order. The subsequent execution design separates history variation from future-input variation. Sixteen histories combine two association orientations, four new notes and issue-time pairs (−6,−1)(-6,-1) and (−5,−3)(-5,-3). Each receives all twenty-four continuations formed by value pairs (3,11),(11,3),(1,13),(13,1),(0,0),(8,8)(3,11),(11,3),(1,13),(13,1),(0,0),(8,8), both sensor arrival orders at times 1,21,2, and both queries. Thus the complete numerical roster behind main-text Table 4 gives 384 base executions. A separate reconstruction reads the recorded protocol, discovery and subsequent comparisons, requests, returns, predictions and selected representation without importing the program, state extractor or predictor. It checks complete input sets, associations, selection, matched pairs and controls; corruption tests alter outputs, row coverage, representations and control baselines. Boundary tests cover negative and fractional values, threshold equality and returns at the horizon. These check the known implementation, not an unknown system. S4.2 Language-model sampling and comparison These comparisons test whether summaries of request histories preserve a model’s later quantity reports. Candidates retain a snapshot (zero current quantity and two pending requests), that snapshot with issue times or sensor–recipient associations, or full history. Matched candidate states must give output laws within the declared tolerance; the descriptions never enter the model’s input. Constructed requests associate sensors with recipients; returns carry the matching request identifier, value and arrival time. The identifiers link returns to requests; their spelling has no task-defined meaning. Inputs also contain an unrelated text note, which the prompt states is not an instruction. A continuation specifies the returns and query. Qwen2.5-14B-Instruct 42 (MLX 4-bit) reads complete records and reports a single-field JSON integer; other completed responses are INVALID. Request–return matching supplies reference answers for scoring correctness separately from response agreement. Main-text Sections 5.3–5.5 report the broader development failure, the recipient-A restriction chosen afterward, and the subsequent tests of identifiers, request order and revised states. Sampling settings and assumptions. The restricted and crossed comparisons request temperature 0.70.7, top-p 11, top-k 00 and a distinct fixed seed per scientific call. The model prompt, record schema, model and parser are identical across the comparisons; the new comparison in Section S4.2.1 changes the joint seed schedule. Sixteen instrument-check calls vary omitted, distinct and repeated seeds and zero temperature. All return the same output and receive no correctness score, leaving effective seed and temperature handling unverified. Development queries A with sensor values (2,7)(2,7), S1S_1 first, or B with (7,2)(7,2), S2S_2 first. The broader development task queries both recipients. For a continuation querying B, two association-matched histories should both give 2: one instead gives 7 in all four calls, the other gives 2 three times and 7 once. The remaining histories give their reference 7 in all four calls each. Thus 5/16 responses are incorrect and the matched pair has empirical total variation (TV) 0.75, the largest event-probability difference computed from the observed frequencies. This motivates the later recipient-A restriction. Histories, comparisons and repeated responses. An input condition fixes one history and one continuation. A matched comparison pairs distinct histories with the same candidate state under a common logical continuation; several comparisons may reuse the same responses. All 16 instrument-check calls and 32 development calls are retained; the restricted selection reuses 16 of the latter, adding no new observations. The restricted follow-up combines both association orientations with two identifier/time/note preparations. Its continuations query A with values (3,11)(3,11), S2S_2 first, or (13,1)(13,1), S1S_1 first. For example, the two histories containing the literal note silver-folder have sensor-ordered issue times (−6,−1)(-6,-1); values (3,11)(3,11) arrive at times (2,1)(2,1). Exchanging associations changes the reference from 3 to 11 at a fixed snapshot. Four histories under two continuation combinations give eight base input conditions for comparing histories at the same candidate state. Four additional control conditions increment queried or unqueried sensor values to test response and invariance, respectively. At 64 calls per condition, the follow-up comprises 512 base and 256 control calls. Repeated calls do not add task cases or establish sampling independence. Original requests, responses and parsing retain every completed output, including the additional invalid category specified in the main text. Reconstruction scope. Verification reads original requests and response bytes without new inference. Local metadata report the model and requested sampling settings; they do not attest model weights, server sampling or independent execution origin. Hashes bind request and representation consistency, without authenticating external registration or chronology. Crossed identifiers, associations and request order. This comparison asks whether the sensor–recipient association state still preserves responses when request identifiers and list order vary independently. The model continues to read full records and to report recipient A’s quantity. The design in main-text Section 5.4 uses two request-identifier schemes: (q17,q42)( q17, q42) and (m83,m29)( m83, m29), indexed by sensors (S1,S2)(S_1,S_2). The first association orientation assigns S1S_1 to A and S2S_2 to B; the second reverses these assignments. Each identifier scheme occurs under both orientations and both request-list orders. Request issue times are fixed at (−6,−2)(-6,-2) by sensor, and the note is fixed. These assignments remove the earlier correlation of identifier family with association orientation; they do not assume token invariance. Return records are ordered by their supplied arrival times. The first continuation supplies sensor values (5,17)(5,17), with S2S_2 returning first; the second supplies (19,4)(19,4), with S1S_1 returning first. Both query A at horizon three. Renaming requests also renames the corresponding return references. Values and arrival order remain two bundled conditions, rather than a fully crossed pair of factors. Each comparison uses two distinct histories with the same candidate state under a common continuation. The snapshot and sensor-sorted issue-time state each have 56 such comparisons; the sensor-sorted association state has 24, and full history has none. Eight single-factor pairs test each of identifier scheme, association orientation and request order. Queried and unqueried increments use both orientations, the first identifier scheme, the original request order and the first continuation. The 20 conditions are interleaved once per sampling round, with 64 rounds and distinct requested seeds. Each reference quantity is calculated from the request–return associations. Lower bounds on its output probability bound the overlap when the reference quantities agree, or separation when they differ. All comparisons follow from 20 simultaneous one-sided reference-probability bounds, assuming independent draws from a stationary conditional output distribution within each condition. With 64 responses equal to the reference the TV upper and lower limits are approximately 0.089370.08937 and 0.821260.82126; these are resolution calculations, not guaranteed experimental outcomes. The combined support criterion requires TV upper bounds at most 0.150.15 for every association-matched comparison and both unqueried-sensor increments; TV lower bounds must exceed 0.500.50 for all eight comparisons that change only association orientation and both queried-sensor increments. All invalid responses enter the denominator. Missing records or transport failures stop the experiment without replacement and preclude a confirmatory conclusion. The crossed comparison leaves association-state agreement unresolved under the fixed rule. Table S2 gives every condition’s counts: all 1,280 outputs are valid, with 981/1,024 base and 149/256 control responses equal to their references. Each snapshot candidate has 28 rejected, 21 within-margin and seven unresolved pairs; the association state has 21 within-margin and three unresolved pairs. Full history supplies no matched pair. Changing identifiers or request order alone gives seven within-margin and one unresolved pair for each factor; changing association orientation gives seven resolved differences above 0.500.50 and one unresolved pair. These comparisons reuse responses rather than add independent tasks. The controls explain the incomplete support. Both pass in the first orientation; both remain unresolved in the second, whose base response often gives the other sensor’s value. Increasing that unqueried value produces empirical TV 63/6463/64, yet the reference-probability interval remains approximately [0,0.99996][0,0.99996]. The bounds can therefore miss changes among non-reference responses. The table retains the underlying counts, and the fixed decisions are unchanged. A separate parser reconstructs all conditions and bounds from saved requests and responses without the experiment runner or new calls. Table S2: Response sensitivity to request names, sensor assignments and list order. Request identifiers (IDs) match returns to their original requests; their spelling has no prescribed effect on the answer. For example, q17/q42 names the requests from S1S_1 and S2S_2, respectively, even when their list order is reversed. Both ID pairs and value pairs follow this sensor order. The sensor assigned to recipient A determines the reference answer. Each row records 64 model responses; changing names, assignments and list order independently tests which distinctions affect them. For base values (5,17)(5,17), S2S_2 returns first; for (19,4)(19,4), S1S_1 returns first. The four controls use the first continuation, increasing one sensor value by one. Entries in the final column are answer (count); every omitted integer and INVALID has count zero. These counts reconstruct both the original reference-probability bounds and the separate exploratory categorical analysis. Request IDs Sensor for A Request order Values Reference Answers (count) q17/q42 S1S_1 S1,S2S_1,S_2 (5,17)(5,17) 5 5(64)5\;(64) q17/q42 S1S_1 S1,S2S_1,S_2 (19,4)(19,4) 19 19(64)19\;(64) q17/q42 S1S_1 S2,S1S_2,S_1 (5,17)(5,17) 5 5(64)5\;(64) q17/q42 S1S_1 S2,S1S_2,S_1 (19,4)(19,4) 19 19(64)19\;(64) q17/q42 S2S_2 S1,S2S_1,S_2 (5,17)(5,17) 17 5(43), 17(21)5\;(43),\;17\;(21) q17/q42 S2S_2 S1,S2S_1,S_2 (19,4)(19,4) 4 4(64)4\;(64) q17/q42 S2S_2 S2,S1S_2,S_1 (5,17)(5,17) 17 17(64)17\;(64) q17/q42 S2S_2 S2,S1S_2,S_1 (19,4)(19,4) 4 4(64)4\;(64) m83/m29 S1S_1 S1,S2S_1,S_2 (5,17)(5,17) 5 5(64)5\;(64) m83/m29 S1S_1 S1,S2S_1,S_2 (19,4)(19,4) 19 19(64)19\;(64) m83/m29 S1S_1 S2,S1S_2,S_1 (5,17)(5,17) 5 5(64)5\;(64) m83/m29 S1S_1 S2,S1S_2,S_1 (19,4)(19,4) 19 19(64)19\;(64) m83/m29 S2S_2 S1,S2S_1,S_2 (5,17)(5,17) 17 17(64)17\;(64) m83/m29 S2S_2 S1,S2S_1,S_2 (19,4)(19,4) 4 4(64)4\;(64) m83/m29 S2S_2 S2,S1S_2,S_1 (5,17)(5,17) 17 17(64)17\;(64) m83/m29 S2S_2 S2,S1S_2,S_1 (19,4)(19,4) 4 4(64)4\;(64) Queried- and unqueried-sensor controls q17/q42 S1S_1 S1,S2S_1,S_2 (6,17)(6,17) 6 6(64)6\;(64) q17/q42 S1S_1 S1,S2S_1,S_2 (5,18)(5,18) 5 5(64)5\;(64) q17/q42 S2S_2 S1,S2S_1,S_2 (5,18)(5,18) 18 5(44), 18(20)5\;(44),\;18\;(20) q17/q42 S2S_2 S1,S2S_1,S_2 (6,17)(6,17) 17 6(63), 17(1)6\;(63),\;17\;(1) Exploratory categorical comparison. Reference-probability bounds can miss changes among incorrect responses. A post-hoc categorical reanalysis resolves differences in the three association-matched comparisons left unresolved, using the scientific-evidence comparison’s simultaneous component intervals with a different output alphabet. The six quantities appearing in the recorded returns supplied to the model across all 20 conditions are 4,5,6,17,18,194,5,6,17,18,19. Each is a category; other valid integers form a seventh category and invalid outputs an eighth. This rule depends on the supplied inputs, not the observed outputs or reference answers. All categories, including zero-count categories, and all conditions enter the uncertainty calculation. Each component receives a two-sided Clopper–Pearson interval with per-tail error η=0.05/(2⋅20⋅8)=0.00015625η=0.05/(2· 20· 8)=0.00015625. Bonferroni control gives nominal simultaneous coverage at least 95% for this specified family under stationary independent sampling within each condition. The method was chosen after the outcomes; this family is separate from the original reference-probability analysis and does not give joint 95% coverage across analyses. Requiring category probabilities to lie in their intervals and sum to one bounds every category-subset probability, and hence TV, as in Section S6. For those three association-matched pairs, the event “quantity equals 5” already attains the resulting lower bound. Its counts are 43/64 in one condition and 0/64 in the other. Writing Bq−1(a,b)B_q^-1(a,b) for the beta-distribution quantile, the separation lower bound is Bη−1(43,22)−B1−η−1(1,64)≃0.3121972.B_η^-1(43,22)-B_1-η^-1(1,64) 0.3121972. The other 21 association pairs have lower bound zero; this analysis makes no positive adequacy classification. Category grouping cannot increase TV, so the lower bound transfers to the original integer/invalid laws; a grouped upper bound would not establish their agreement. A state-only predictor must use one law for both inputs. The half-separation bound (main-text Section 8.3) therefore gives worst-case behavioral error at least 0.15600.1560, not a task error rate. All 24 pairs and original decisions remain reported; the reanalysis adds no calls. S4.2.1 Testing added identifiers and request order The new comparison tests whether request identifiers and list order are needed in addition to sensor–recipient associations on new recipient-A inputs. The analyst compares the simpler and revised states; the model’s task and full-record inputs remain fixed. Main-text Section 5.5 reports the results. All requests, candidates, margins, sample sizes and analysis rules were fixed before its responses. No observations from the earlier studies enter its bounds. The model, prompt, strict integer/invalid parser, temperature 0.70.7, top-p 11, top-k 00, and 96-token output limit are unchanged. Inputs and candidate states. The revision retains the snapshot and ordered (request ID,sensor,recipient)(request ID,sensor,recipient) triples. Two further candidates retain order without IDs, or IDs without order. Two arbitrary request-ID pairs, (v36,v91)( v36, v91) and (r64,r28)( r64, r28), link sensors (S1,S2)(S_1,S_2) to their returns. Each pair occurs under both association orientations and request orders. To create distinct histories with the same revised state, two preparations jointly vary request times and the unrelated input note, both omitted by that state. They use issue times (−9,−3)(-9,-3) with the literal text grey-folder, or (−2,−7)(-2,-7) with orange-folder; these are note contents, not file references. Both requests precede checkpoint zero; relative issue chronology reverses. Timing and note vary together, so their separate effects are not identified. The two continuations bundle values (6,23)(6,23) and (27,8)(27,8) with opposite arrival orders, both querying A at T=3T=3. Each comparison matches two distinct histories at the same candidate state under a common continuation. Their counts are 240 for the snapshot, 112 for associations, 48 for each partial extension, 16 for the revision and zero for complete history. Forty-eight of the association comparisons have identical raw future returns; the other 64 consistently relabel return references. All primary revision pairs hold raw future returns identical. Four controls increment the queried or unqueried sensor in each orientation, using the first ID scheme, original request order, first timing/note setting and first continuation. They cover these layouts only. Reference-free simultaneous bounds. These bounds compare complete output laws using category frequencies for separation and shared probability mass, and paired-output disagreement for an additional agreement bound. The inputs above form 32 base conditions and four controls, giving 36 conditions in total. Each of 256 replicates requests one common seed for all 36 conditions; replicates use distinct seeds and fixed interleaved condition orders. Inference assumes independent stationary replicate vectors with the declared marginals, allowing dependence within a replicate. Requested seeds do not attest these assumptions or their implementation. Let =6,7,8,23,24,27,INVALIDS=\6,7,8,23,24,27,INVALID\ denote the seven individual parsed outcomes fixed from the inputs and parser. Each is kept separate; all other valid integers are grouped into one residual category for computation only. Allocate error 0.0250.025 across both tails of all 36×836× 8 component intervals, using per-tail error η=0.025/(2⋅36⋅8)η=0.025/(2· 36· 8). Requiring category probabilities to lie in these intervals and sum to one gives lower and upper event-probability bounds ℓi(E),ui(E) _i(E),u_i(E), as in Section S6. Over subsets of the eight categories, d¯ij=maxE0,ℓi(E)−uj(E),ℓj(E)−ui(E) d_ij= _E\0, _i(E)-u_j(E), _j(E)-u_i(E)\ lower-bounds TV of the original integer/invalid laws. For agreement, the unresolved integers in the residual category cannot be counted as shared output. Overlap in the individual categories alone gives the conservative upper bound d¯ijmarg d^\,marg_ij, where margmarg denotes the marginal output distributions: d¯ijmarg=1−minE⊆ℓi(E)+ℓj(∖E). d^\,marg_ij=1- _E \ _i(E)+ _j(S E)\. Write YiY_i for the full parsed output in condition i, with probability law Πi _i. Under any joint sampling with these marginal laws, dTV(Πi,Πj)≤Pr(Yi≠Yj)d_TV( _i, _j)≤ (Y_i≠ Y_j). The reported pair family contains all snapshot-matched comparisons and the four controls; other candidate comparisons are subsets of the base family. Count unequal full parsed outputs in each paired replicate, including INVALID and retaining distinct residual integers. One-sided exact binomial upper bounds allocate the remaining 0.0250.025 error across all 244 distinct reported pairs. The minimum of this upper bound and d¯ijmarg d^\,marg_ij, together with d¯ij d_ij, has nominal simultaneous coverage at least 95% under the sampling assumptions. Pairwise independence is unnecessary. Reference quantities enter only the separate correctness score. Decisions and complete outcomes. The fixed margins are TV 0.150.15 for agreement and 0.500.50 for positive controls. The primary refinement criterion requires both rejection of the association-only description on the new inputs and agreement for all revised pairs; the combined criterion also requires the four controls. Complete history is uninformative. No retry, replacement, outcome-dependent extension or deletion is permitted; all 9,216 calls complete with valid outputs. Both revised and association-only states have bounded agreement, and the controls pass. The primary refinement criterion fails because the simpler state is not rejected. Agreement on these inputs leaves the earlier exploratory separations unchanged. The snapshot has 128 rejections and 112 within-margin comparisons; all other nontrivial candidates are within margin on their full declared comparisons. Of the 16 revised-state pairs, 15 have no paired output mismatch and one has 2/256 mismatches, giving upper bounds 0.03530.0353 and 0.05310.0531, respectively. The 112 association pairs have these bounds in 105 and seven pairs; both partial extensions have 45 and three such pairs. Snapshot rejection lower bounds exceed 0.90480.9048. The two positive-control lower bounds exceed 0.92300.9230; both invariance-control upper bounds are at most 0.03530.0353. The only two wrong responses occur with the second ID scheme, S2S_2 assigned to A, original request order, second timing/note setting and first continuation: 254 responses give 23 and two give the other sensor’s value 6. All other cells have 256 correct responses. The authors retain requests and raw responses. A separately implemented parser and analysis reproduce every count, bound and classification without the experiment runner or new inference. This checks the recorded results, not the server’s implementation of the requested sampling settings. S5 Methods for inverse tests under tool feedback This instructed workflow tests whether an initial summary must retain software version information to predict the next check and release decision. Release requires both unit and integration checks to pass for the current version. Each history has two focal PASS results: one current and one predecessor result, for different check types. Exchanging their version assignments changes which current check is missing while preserving both statuses. Confirmation adds a redundant predecessor unit-check result. Requests identify version and check type; returns identify their requests. The experiment program joins them before Qwen2.5-14B-Instruct (MLX 4-bit) reads the completed results and outstanding requests. Model decisions use fresh calls; histories, checks, release and hold are constructed or simulated, without generated-code execution or external deployment. Main-text Section 6 reports the comparisons. The instructions below specify the complete decision rule. The model reports unit status, integration status, then action, in that order. The schema allows every combination of PASS, FAIL, PENDING or ABSENT with the five actions; it supplies no assessed status or correct action. Requested temperature is 0.70.7, top-p 11, top-k 00, with an 80-token completion limit and distinct recorded seeds. Duplicate, missing, extra or misordered fields, disallowed values, malformed JSON and unfinished completions are invalid. Recorded server metadata identify the loaded model, without cryptographic verification of its weights. The verbatim instruction prescribes the status assessment and identifier/order invariance; the current-version candidate retains this explicitly required information. Underscores identify transmitted JSON fields. Schedule validation of the current software artifact. Both unit and integration checks must PASS on this exact artifact before release. completed_results lists finished checks with their artifact, check type and status. pending_requests lists checks requested but not returned. These lists are complete. An old-artifact check is neither a completed nor pending check for the current artifact. First assess unit_status and integration_status for the CURRENT artifact: PASS or FAIL for its completed check; PENDING for its outstanding request; ABSENT if neither exists. Then choose the next action. If either current check is FAIL, HOLD. If both are PASS, RELEASE. If a needed current check is PENDING, WAIT. Otherwise RUN_UNIT or RUN_INTEGRATION for an ABSENT check. Return JSON in this exact field order: unit_status first, integration_status second, action last. Record order and identifier spelling do not change which artifact was checked. The continuation fixes the instructed action rule, simulated check outcome and whether the return is delivered or withheld (the observation branch). A requested check returns the externally fixed PASS or FAIL for the current version, hidden from the first model call. Two branches reuse that sampled first action: one receives the executed request and return; the other receives the request with its return withheld. Fresh second calls use the same instructions, in an order fixed by a shuffled schedule. A first action other than requesting a test terminates the episode. The measured sequence therefore contains at most two decisions. Its sixteen categories are each of the two test-request actions followed by any of the five actions or INVALID, together with four sequences ending at the first decision: WAIT, RELEASE, HOLD and INVALID. A second test request is recorded but does not extend the window. Status assessments and arbitrary generated text are not part of this readout. Three predefined, analyst-only summaries retain respectively the two summarized statuses and pending-request count; the latest status for each check without version information; or unit and integration results for the current version, including absence. They never replace the model’s records. The first two are constant and induce the same partition on the tested initial histories; their repeated rejection does not represent independent discoveries. The third omits identifiers, row order and redundant predecessor results while separating the two missing-check situations. These initial preparation summaries omit later pending flags, so agreement does not test autonomous state evolution. Automatic discovery and uniqueness of the summary are also untested. Interface selection and confirmation. Development screens interfaces for following the supplied decision rule. Two missing-check situations and two return conditions receive eight starting episodes each. The fixed screen requires at least seven correct first choices and seven correct feedback sequences in every condition. Action-only output gives RELEASE on all 32 calls. A separate presentation test (32 calls) finds no format meeting both missing-check requirements. Joined records with status assessment give two correct first choices in 32 episodes (36 calls): alphabetic schema ordering requests action before assessment. Preserving the intended field order instead gives 30 correct choices and feedback sequences in 32 episodes (92 calls), with counts 8,8,7,78,8,7,7, passing the screen. Confirmation retains this interface and its inference and analysis code; correcting an empty-input test fixture changes neither. Development responses fit forecasts but do not enter confirmation counts. Confirmation crosses two missing-check situations, two identifier schemes and two request-list orders, giving eight histories, each under PASS and FAIL: 16 conditions with 64 starting episodes each. New version names, request identifiers and the redundant predecessor result distinguish these inputs from development; the two version names are fixed within confirmation, so version-name invariance is not tested. Within each sampling round every condition occurs once, in a fixed shuffled order; branch order is also fixed before execution. The local record binds the source revision, analysis files, development forecasts, model configuration and sampling limit before the recorded calls. This is a local execution record, not independent registration or external attestation of chronology. All 1,024 episodes and 3,072 calls completed. These repeats do not constitute 1,024 independently sampled tasks. Invalid outputs remain in every denominator; transport failure stops execution without retry, incomplete confirmation permits no acceptance, and outcomes never trigger additional sampling. Comparing action-sequence distributions. Histories merged by a candidate summary are compared under the same imposed tool outcome. Category-probability intervals bound how much their sequence distributions can differ, including invalid outputs. The 16 history/outcome conditions each have a delivered-return and a withheld-return branch, giving M=32M=32 distributions. Let XikX_ik count sequence category k in distribution i, and pikp_ik denote its probability, with n=64n=64 observations and K=16K=16 categories. Conditional on the fixed development fit and interface, assume independent, stationary repetitions within each distribution. The branches share their first action and are not independent trials; independence across branches, conditions or categories is unnecessary. Seeds and interleaving do not establish these assumptions. For family error α=0.05α=0.05, allocate t=α/(2MK)t=α/(2MK) to every interval tail, including zero-count categories. With Bq−1(a,b)B_q^-1(a,b) the beta-distribution quantile, the Clopper–Pearson bounds are9 Lik=Bt−1(Xik,n−Xik+1),Uik=B1−t−1(Xik+1,n−Xik),L_ik=B_t^-1(X_ik,n-X_ik+1), U_ik=B_1-t^-1(X_ik+1,n-X_ik), (S5.1) using Lik=0L_ik=0 at zero counts and Uik=1U_ik=1 at n. The union bound over all 2MK2MK tails gives joint nominal coverage at least 95% under these assumptions. On this event each true law lies in i=p:∑kpk=1,Lik≤pk≤UikP_i=\p: _kp_k=1,\ L_ik≤ p_k≤ U_ik\. Total variation is dij=12∑k|pik−pjk|d_ij= 12 _k|p_ik-p_jk|. Define the unavoidable coordinate gap gijk=max0,Lik−Ujk,Ljk−Uikg_ijk= \0,L_ik-U_jk,L_jk-U_ik\. The fixed bounds are d¯ij=maxmaxkgijk,12∑kgijk,d¯ij=1−∑kminLik,Ljk. d_ij= \! \ _kg_ijk, 12 _kg_ijk \, d_ij=1- _k \L_ik,L_jk\. (S5.2) The lower bound follows from coordinate differences and the upper bound from guaranteed shared probability mass. Neither uses correctness labels. Bounded agreement requires d¯≤0.15 d≤ 0.15, rejection requires d¯>0.15 d>0.15; otherwise the comparison is unresolved. Identical observed point masses give d¯=1−t1/64≤0.1437 d=1-t^1/64≤ 0.1437; distinct point masses give d¯=2t1/64−1≥0.7126 d=2t^1/64-1≥ 0.7126. For these settings, any dispersed observed histogram prevents agreement at 0.150.15: the smallest attainable upper bound for such a comparison exceeds 0.17910.1791. This limits resolution without invalidating the bounds or proving that all-identical samples arose from deterministic laws. Only histories with equal candidate summaries and the same return condition are paired in the feedback branch: 56 pairs per simpler summary and 24 for current-version results. Primary support requires a resolved violation for each simpler summary under each return condition, agreement on all 24 revised pairs, and lower bounds above 0.500.50 for all eight same-history PASS/FAIL controls. Feedback participation separately requires that threshold for all 16 feedback/withheld-return contrasts. These comparisons reuse observations and the same confidence event. For two merged histories separated by d, any common predicted law has worst-case TV error at least d/2d/2, by the triangle inequality. The observed separation implies at least 0.35630.3563; this is a behavioral approximation bound, not a task-error rate or an exact optimal error. Comparing fitted forecasts. The fitted forecasts compare action-sequence prediction from the status-only and current-version summaries, conditional on the initial summary, known PASS/FAIL forcing and observation branch. Saved before confirmation, they use no sampled future action, assessment or acquired confirmation result; they do not forecast an unknown test outcome. Development sequences in Table S3 are pooled by these inputs. If NgkN_gk counts category k in group g, the fixed forecast is qgk=(Ngk+1/16)/(Ng+1)q_gk=(N_gk+1/16)/(N_g+1), where Ng=∑kNgkN_g= _kN_gk. Both forecasts use a total uniform pseudocount of one: Ng=16N_g=16 for the status-only summary and Ng=8N_g=8 for current-version results. The instructed rule gives a point-mass forecast from the same records and forcing without fitting. Table S3: Development counts used to fit the closed-loop forecasts. Each row contains eight starting episodes. Unit and Integration denote the actions requesting those tests; an arrow gives the second action. Entries are sequence (count), with zero for all omitted categories among the sixteen defined above. WAIT alone terminates at the first decision and is counted in both branches without an additional call. The branches share their first action; the table represents 32 episodes and 92 calls, not 64 independent trials. Pool rows with the same return condition for the status-only forecast; keep the missing-test distinction for the current-version forecast. Missing test Return Feedback sequences Return-withheld sequences Unit PASS Unit → RELEASE (8) Unit → WAIT (8) Unit FAIL Unit → HOLD (8) Unit → WAIT (8) Integration PASS Integration → RELEASE (7); WAIT (1) Integration → WAIT (7); WAIT (1) Integration FAIL Integration → HOLD (7); WAIT (1) Integration → WAIT (7); WAIT (1) Forecast accuracy is measured by the normalized Brier loss7, ℓ(q,k)=12∑j(qj−j=k)2=12(1+‖q‖22)−qk (q,k)= 12 _j(q_j-1_j=k)^2= 12(1+\|q\|_2^2)-q_k. Expected loss is linear in the unknown probabilities. Its extrema over iP_i are obtained by starting at LiL_i and allocating remaining mass up to UiU_i, in increasing or decreasing loss order. Let qis,qivq_i^s,q_i^v denote the status-only and current-version forecasts and ℱF the 16 feedback conditions. Their equally weighted expected loss difference Δ has bounds Δ¯=116∑i∈ℱmin∑kp∈ipk[ℓ(qis,k)−ℓ(qiv,k)],Δ¯ replaces min by max. = 116 _i _p _i _kp_k[ (q_i^s,k)- (q_i^v,k)], replaces min by max. (S5.3) Optimizing the difference retains the common outcome law for both forecasts. The fixed prediction criterion is Δ¯>0 >0. The fitted current-version forecast meets this criterion: the observed mean loss difference is approximately 0.23780.2378, and the conservatively rounded lower bound on the expected difference is Δ≥0.1180 ≥ 0.1180. The corresponding outward-rounded expected-risk intervals are [0.2517,0.3194][0.2517,0.3194], [0.0181,0.1379][0.0181,0.1379] and [0,0.1437][0,0.1437] for status-only, current-version and instructed-rule forecasts, respectively; the rule has zero observed loss. The loss-difference interval is [0.1180,0.2379][0.1180,0.2379]. All these bounds share the original confidence event, conditional on the fitted forecasts, and do not evaluate repeated refitting on new development samples. Comparisons of these fitted forecasts establish neither each summary’s optimal loss nor superiority over the instructed rule. In confirmation every observed first action is identical within its condition. The withholding comparison therefore tests one subsequent decision for that action, not robustness across alternative first actions or longer feedback sequences. A summary can preserve the final decision while omitting how it was reached. A supplementary analysis chosen after interim observations tests each sequence’s last action, including INVALID. Minimizing and maximizing each terminal-action indicator over the original iP_i supplies component bounds for the projected laws; applying the same TV formulas yields conservative comparisons. All 56 status-only pairs then have bounded agreement, whereas 32 reject for full sequences; PASS/FAIL controls remain resolved. This projection adds neither observations nor an independent confidence guarantee, and leaves the fixed primary classifications unchanged. A terminal action is not a terminal state, and agreement at that readout does not validate the initial summary for intermediate actions or autonomous dynamics. Separate pilots test whether feedback improves task outcomes. In one, a controller adjusts capacity over four decisions to regulate the difference between demand and capacity. Its free-JSON attempt stopped at request 11 after an unfinished response; a separate structured-output run completed 104 calls. Returned observations changed trajectories, but all four nonzero pulses failed the regulation criterion; two of five evaluation amplitudes exceeded the fixed scalar-prediction error limit. A second pilot tests feedback-guided acquisition of scientific evidence. Its 216 calls reuse eight SciFact training claims in sixteen excerpt conditions, once per arm. Feedback and frozen observations each give 11/16 final source-label agreements, versus 12/16 for retrieval-order acquisition with fewer calls. These are descriptive source-label comparisons, not adjudicated accuracy for modified inputs. Neither pilot supplies task-benefit evidence or enters this confidence family. The two-check experiment excludes dependency-graph faults, cancellation, retries and interacting agents; it does not validate the broader stopped study in Section S7.5. S6 Methods for scientific-text comparisons This experiment asks whether a document list or an unordered set of excerpts preserves a reader’s responses when the selected text or its document order changes. The reader receives a scientific claim and excerpts, then judges support, refutation or insufficient evidence. Repeated judgments estimate output distributions; comparing these distributions tests the candidate representations, separately from answer correctness. Inputs and comparisons. SciFact associates scientific claims with research abstracts and annotated evidence sentences57. The design uses the first eight entries of a fixed, ordered list of 96 claims from its official training split, previously used in the action-selection development study (89 of 96 tasks completed; Section S7.4), without substitution or screening by model outputs. Claim-only BM25 retrieval, a keyword-based ranking procedure, supplies six abstracts per claim, with term-frequency saturation parameter k1=1.2k_1=1.2 and document-length normalization parameter b=0.75b=0.75. It uses Lucene inverse-document-frequency weighting and document-identifier tie-breaking. Selecting the first or last three sentences of each abstract, then retaining or reversing document order, gives M=8×2×2=32M=8× 2× 2=32 input conditions (cells), each containing 18 sentences. Sentence order within each abstract is preserved. Leading and trailing excerpts overlap in seven of the 48 retrieved abstract occurrences; selection policies do not imply disjoint text. Two analyst-defined states retain either the claim, document identities and excerpt counts, or the claim and unordered exact excerpts with titles, document identities and sentence indices. A matched pair consists of two input conditions with the same candidate state. The document-level state matches all pairs of the four conditions; the excerpt state matches the two document orders within each excerpt policy. These give six and two matched pairs per claim, respectively. Two excerpt-policy contrasts at fixed document order provide positive controls for detecting a response. The reader receives ordered text, without candidate states or annotations. Annotations measure complete-rationale retention only; missing annotated evidence neither defines an insufficient-evidence answer nor supplies a correctness target for shortened inputs. Sampling and simultaneous intervals. Input conditions, sampling settings and comparison rules were fixed before the calls and checked against local hashes during execution. These records document local preparation and integrity, without independent timestamp attestation. The fixed Qwen2.5-14B-Instruct reader (MLX 4-bit) received 64 calls per cell, interleaved across cells: 2,048 calls, not additional claims. Requested temperature is 0.70.7, top-p 11, top-k 00, with a 16-token output limit. After removing surrounding whitespace, exact A/B/C completions encode support, refutation and insufficient evidence; all other completions are invalid. Thus the alphabet is =SUPPORTS,REFUTES,NEI,INVALIDY=\SUPPORTS,REFUTES,NEI,INVALID\, with K=4K=4. Let XikX_ik count category k in cell i, ni=∑kXikn_i= _kX_ik, and pikp_ik its probability. Full counts, including invalid responses, enter every comparison without using reference answers. Transport or record-integrity failure halts sampling without retry; incomplete runs yield no adequacy conclusion, and thresholds never trigger extra draws. The intervals quantify uncertainty in each response-category probability before those probabilities are compared across matched inputs. Assume fixed sample sizes and independent stationary repeats within each cell; requested seeds establish neither property. For family error α=0.05α=0.05, set the per-tail error budget to t=α/(2MK)t=α/(2MK). Writing Bq−1(a,b)B_q^-1(a,b) for the q-quantile of the beta distribution with shape parameters a,ba,b, the Clopper–Pearson intervals [Lik,Uik][L_ik,U_ik]9 are Lik=Bt−1(Xik,ni−Xik+1),Uik=B1−t−1(Xik+1,ni−Xik).L_ik=B_t^-1(X_ik,n_i-X_ik+1), U_ik=B_1-t^-1(X_ik+1,n_i-X_ik). (S6.1) Use Lik=0L_ik=0 when Xik=0X_ik=0, Uik=1U_ik=1 when Xik=niX_ik=n_i, and [0,1][0,1] when ni=0n_i=0. Each tail fails with probability at most t; the union bound over 2MK2MK tails gives simultaneous coverage at least 1−α=0.951-α=0.95, without independence across cells or categories. Event and distance bounds. A response event is any subset of output categories, such as support or refutation. To test whether matched inputs have similar output laws, bound the largest probability difference over all such events. On the simultaneous coverage event above, each category probability lies in its interval and the probabilities sum to one. For every nonempty proper subset E⊂E , with complement EcE^c, its probability extrema are ℓi(E)=max∑k∈ELik,1−∑k∈EcUik,ui(E)=min∑k∈EUik,1−∑k∈EcLik. _i(E)= \! \ _k∈ EL_ik,1- _k∈ E^cU_ik \, u_i(E)= \! \ _k∈ EU_ik,1- _k∈ E^cL_ik \. (S6.2) Indeed, the attainable sum intervals on E and its complement intersect at exactly these endpoints after normalization. Total variation, dij=12∑k|pik−pjk|=maxE[pi(E)−pj(E)]d_ij= 12 _k|p_ik-p_jk|= _E[p_i(E)-p_j(E)], has simultaneous lower and upper bounds d¯ij d_ij and d¯ij d_ij: d¯ij d_ij =max0,maxE[ℓi(E)−uj(E)],maxE[ℓj(E)−ui(E)], = \! \0, _E[ _i(E)-u_j(E)], _E[ _j(E)-u_i(E)] \, (S6.3) d¯ij d_ij =min1,maxE[ui(E)−ℓj(E)],d¯ij≤dij≤d¯ij. = \! \1, _E[u_i(E)- _j(E)] \, d_ij≤ d_ij≤ d_ij. Each event difference is bounded separately; all 14 subsets include both directions through complements. All pair bounds share the same coverage event, requiring no further multiplicity correction. An unsampled cell gives a conservative pair interval [0,1][0,1]. For two samples concentrated on the same category, 64 draws per cell give d¯=1−t1/64≤0.1250 d=1-t^1/64≤ 0.1250, below the 0.150.15 agreement margin. Concentration on distinct categories gives d¯=2t1/64−1≃0.7501 d=2t^1/64-1 0.7501. These precision cases motivate the fixed repeat count without guaranteeing a resolved comparison. For this sample size, alphabet and simultaneous error allocation, agreement can be certified only for two observed samples concentrated on the same category. Any dispersed observed histogram makes the upper bound exceed 0.150.15; this restriction concerns the samples and comparison rule, not determinism of the underlying laws. An unresolved interval can also accompany a substantial empirical difference. Its overlap with the margin therefore supports neither agreement nor rejection; increasing sample size would improve precision, not predetermine either conclusion. Reporting and scope. An upper bound at most 0.150.15 gives bounded agreement; a lower bound above 0.150.15 rejects that tolerance; otherwise the comparison is unresolved. Informative adequacy for an excerpt-set comparison additionally requires Ui,INVALID≤0.15U_i,INVALID≤ 0.15 at both matched cells and at both cells of at least one same-claim policy control with d¯>0.5 d>0.5. These reporting conditions reuse the simultaneous intervals; they do not filter responses or stop calls. Failed validity bounds withhold positive adequacy support. Resolved violations of the four-category tolerance remain reportable, including differences in format failure; they do not by themselves isolate a semantic cause. Identical invalid outputs cannot supply adequacy support; identical wrong valid labels can supply bounded agreement. Approximate agreement is not transitive and does not define equivalence classes (main-text Section 8). Conclusions concern output distributions for these claims, not correctness, held-out performance, automatic state discovery or general sufficiency. Complete observations. Both state descriptions have resolved violations on the full tested claim set under the stated sampling assumptions. All planned calls completed; a separate calculation reproduced every count, comparison and resource total. Table S4 retains all responses: 2,048 repeated measurements of eight claims. Table S4: All scientific-evidence response counts. The reader judges a claim from six retrieved abstracts after excerpt selection and document ordering. Columns give these input conditions; tuple entries count model judgments, not correct answers. Each tuple is ordered (SUPPORTS, REFUTES, NEI, INVALID) and sums to n=64n=64. NEI denotes insufficient evidence; INVALID retains all other completions. Leading and trailing select the first and last three sentences per abstract; retrieval and reversed denote document order. Claim wording is reproduced as input, with its SciFact source identifier in parentheses, rather than as a scientific conclusion. In the quoted inputs, Ab denotes antibody and eRNAs denotes enhancer RNAs. The simultaneous invalid-probability upper bound is at most 0.12500.1250 in every cell. Leading excerpts Trailing excerpts Source claim (identifier) Retrieval Reversed Retrieval Reversed Depletion of T-helper 17 (Th17) cells during simian immunodeficiency virus (SIV) infection decreases dissemination of Salmonella Typhimurium from the gut. (332) 64,0,0,064,0,0,0 64,0,0,064,0,0,0 64,0,0,064,0,0,0 64,0,0,064,0,0,0 RAD52 is involved in break-induced DNA replication (BIR). (1003) 64,0,0,064,0,0,0 64,0,0,064,0,0,0 62,0,2,062,0,2,0 32,2,30,032,2,30,0 Exhaustion of B cells contributes to poor Ab response in HIV-infected individuals. (398) 0,0,64,00,0,64,0 0,0,64,00,0,64,0 0,1,63,00,1,63,0 0,0,64,00,0,64,0 40mg/day dosage of folic acid and 2mg/day dosage of vitamin B12 does not affect chronic kidney disease (CKD) progression. (12) 0,0,64,00,0,64,0 0,0,64,00,0,64,0 64,0,0,064,0,0,0 0,12,52,00,12,52,0 Binding of p53 to enhancer regions (p53BERs) regulates the activity of p53 transcription and cell cycle arrest through the production of p53-dependent eRNAs. (177) 62,0,2,062,0,2,0 58,2,4,058,2,4,0 64,0,0,064,0,0,0 64,0,0,064,0,0,0 Medications to treat obesity have unwanted side effects. (766) 63,1,0,063,1,0,0 64,0,0,064,0,0,0 64,0,0,064,0,0,0 64,0,0,064,0,0,0 Acute ablation of Snail in the embryonic cortex affects the proliferation and number of embryonic cortical precursors. (81) 10,3,51,010,3,51,0 0,0,64,00,0,64,0 63,1,0,063,1,0,0 64,0,0,064,0,0,0 taa1 tar1 tar2 triple mutants in Arabidopsis do not make root meristem during embryogenesis. (1405) 0,0,64,00,0,64,0 0,0,64,00,0,64,0 0,0,64,00,0,64,0 0,0,64,00,0,64,0 Across the 48 pairs matched by claim, document identities and excerpt counts, 21 have bounded agreement, nine reject the 0.150.15 tolerance and 18 remain unresolved. The 16 pairs also matched by exact unordered excerpts are a subset: nine have bounded agreement, one rejects and six remain unresolved. Of the remaining pairs, 16 change only excerpt policy and 16 change both policy and order; each group has six bounded, four rejected and six unresolved comparisons. The two rejections for claim 1003 have lower bound 0.15780.1578, slightly above the fixed 0.150.15 tolerance. Only the leading-excerpt pair for claim 12 additionally satisfies the validity and positive-control conditions. Two of the 16 fixed-order policy controls exceed the 0.50.5 lower-bound threshold: claim 12 in retrieval order and claim 81 in reversed order. The trailing-excerpt pair for claim 12 has d¯≃0.75014 d 0.75014. Every packet for claim 332 retains a complete annotated contradicting rationale, yet all 256 responses are SUPPORTS. Their agreement lacks a detectable positive control; the annotation disagreement is not an adjudicated accuracy score (main-text Section 7). The comparisons concern these eight reused claims. S7 Graph response methods and additional workflow studies These studies ask three distinct questions: whether a prescribed graph response is calculated correctly, whether response measurements diagnose an output error, and whether an intervention improves the task outcome. The sections separate deterministic checks, fresh model measurements and exploratory interventions. Constructed cases do not estimate deployment failure rates; repeated calls do not add tasks. Section S8.1 summarizes reconstruction coverage. S7.1 Prescribed graph responses and model readouts Response operators and numerical checks. The numerical checks ask whether the specified dynamics transmit a source change into the predicted state and readout changes. Table S1 defines the local source operators; Section S2.5 defines the graph response and derivative conventions used in these checks. The transition A and source map B follow from declared dynamics or finite source perturbations, not endpoint fitting. Consensus includes delay histories. The code-repair graph uses endpoint information; its readout consumes scalar values and type labels, and sources are separated by site and channel. Minimality, sufficiency and internal model-state dimension require separate tests. For linearized graph dynamics closed in the retained coordinates, with time-independent maps A,BA,B and ρ(A)<1ρ(A)<1, Section S2.2 gives the resolvent. Its residual checks the solve for assembled matrices, not matrix construction. Sections S1.6 and S3.1 define the spectral and consensus participation ratios, respectively, including the latter’s modal-weight bounds and quadrature normalization. Checking response composition in a prescribed graph. The deterministic check asks whether direct input perturbations and composed source-to-state/readout derivatives give the same action-score response. Two small programming tasks—summing even numbers and testing strict sequence increase—each have four candidate repair scores. Expected repairs set the labels, source destinations and some weights, so agreement tests a prescribed calculation, not independent repair performance or model dynamics. The linear graph has ρ(A)=0.941<1ρ(A)=0.941<1, allowing the resolvent (I−A)−1(I-A)^-1. Its solve residual is ‖(I−A)χ∗G−B‖∞=1.55×10−15\|(I-A) _ ^G-B\|_∞=1.55× 10^-15; finite-difference responses and finite-window transition products agree to maximum absolute error 1.65×10−131.65× 10^-13. These check different calculations of the specified graph response. For four action scores s(z)s(z), distinct from augmented state s(t)s(t), and expected repair a∗a_ , differentiable endpoint response uses soft(z)=softmax(s(z)),qsoft(z)=[soft(z)]a∗, q_ soft(z)=softmax\! (s(z) ), q_ soft(z)= [ q_ soft(z) ]_a_ , The binary readout has no differential at its jump. An action is recommended only when its maximum score reaches threshold θ; otherwise none is emitted. Thus PfP_f indicates an allowed recommendation and Qf=[arec=a∗]≤PfQ_f=1[a_ rec=a_ ]≤ P_f. Raw-score margins locate these boundaries: mP(z)=maxasa(z)−θ,mQ(z)=sa∗(z)−maxa≠a∗sa(z),m_P(z)= _as_a(z)-θ, m_Q(z)=s_a_ (z)- _a≠ a_ s_a(z), Away from ties, Pf=1P_f=1 iff mP≥0m_P≥ 0, and Qf=1Q_f=1 iff also mQ>0m_Q>0; ties use recorded action order. The softmax differential measures response, whereas margins classify binary outcomes. Four event amplitudes per task supply source directions through embedding EsrcE_ src into an 88-site, 44-channel graph. Comparing their four output-probability responses with DzsoftAt−t′BEsrcD_z q_ softA^t-t BE_ src gives maximum absolute error 1.13×10−81.13× 10^-8 at amplitude step 0.0030.003. Increasing the step to 0.30.3 changes estimates by at most 1.10×10−31.10× 10^-3 relative to the baseline. Raw-score differences also agree with the composed derivatives. Zeroing propagation and cyclically permuting endpoints give relative errors 1.00 and 1.63, respectively. These deliberate controls check use of the maps; both tasks reuse one topology and are not independent graph replications. Replay of events from recorded model calls. Recorded model calls test whether visible records can define source events without consulting hidden answers. Seven file-grounded tasks from a live Qwen2.5-14B-Instruct-MLX run each supply a plan, tool result and answer candidate; six candidates are correct. Missing verification events are not synthesized. The readout compares each visible answer candidate with a no-answer option. Recorded statuses and a fixed candidate-event rule set source amplitudes. Hidden answers enter only endpoint scoring; the incorrect task’s true answer is absent from its actions and sources. The three event classes have amplitudes (0.7,1.0,1.0)(0.7,1.0,1.0) in every task, so all 21 events instantiate one numerical response template. Its stored source-to-state matrix has rank three; normalization limits the nonzero two-action probability response to rank one. The response vanishes before source time, and difference-amplitude estimates vary by at most 9.71×10−149.71× 10^-14. These deterministic replay checks establish source encoding and propagation, not fresh model response or internal dynamics. S7.1.1 Fresh model readouts after imposed propagation This experiment tests how changes in evidence amplitude affect a model’s support for candidate answers after prescribed graph propagation. Each of seven constructed tasks requires choosing one of three values for each of three categorical output variables. Main Section 4 specifies the task records, event extraction and checks, graph construction, state update and candidate readout in execution order. Section S1.4 defines the general source-encoding map. Here the sampling procedure separates development choices, fixed-graph perturbations and readout controls; hidden endpoint verifiers score the model’s responses separately. Each extraction block consists of one instance evaluated with one requested seed. Three instances/two requested seeds form development; four other instances/one seed form the confirmatory measurement, excluded from protocol calibration. Development selects load λ=0.9375λ=0.9375, rather than calibrating a natural boundary. Candidate input is λ times within-field mean support plus 1−λ1-λ times candidate support, attenuating contrast by 1616. The pilot fixes state-distance threshold 0.050.05 and absolute candidate-slope threshold 0.010.01 before confirmation. Externally constructed graph coordinates are not internal model states. The measured graph follows main Eq. (19) with A=αI+βWA=α I+β W, where α and β are dimensionless uniform retention and transport gains and W specifies transport between graph coordinates. The recorded settings are α=0.72α=0.72 and β=0.45β=0.45. Four updates deposit each event at its assigned time, apply A, rescale if the maximum exceeds one, and clip to [0,1][0,1]. Neither rescaling nor clipping fires on the reported 4848 perturbed trajectories (192192 updates): the largest coordinate before rescaling is 0.56510.5651 and none is negative, so the reported perturbations lie in the linear regime of the propagator. For this graph, W2=0W^2=0: edges run only from event coordinates to terminal candidate coordinates. The binomial expansion gives, for integer n≥1n≥ 1, An=αnI+nαn−1βW.A^n=α^nI+nα^n-1β W. An amplitude contrast deposited at td<Tt_d<T therefore gives Δzd(T)=AT−tdedΔud z_d(T)=A^T-t_de_d\, u_d, provided range control acts as the identity along both trajectories. This is an exact graph-state response; the model-readout measurements concern the fixed coefficient pair above. Baseline event amplitude is 0.50.5. Selected direction d changes only one event by ±ϵ±ε, with ϵ=0.1ε=0.1 and refinement ϵ/2=0.05ε/2=0.05. Scalar score s=OTs=O_T from main Eq. (21), distinct from full state s(t)s(t), gives Δzd(T)=zd,+(T)−zd,−(T),g^ds(ϵ)=sd,+−sd,−2ϵ. z_d(T)=z_d,+(T)-z_d,-(T), g^s_d(ε)= s_d,+-s_d,-2ε. Distance ‖Δzd(T)‖2\| z_d(T)\|_2 uses every terminal graph coordinate. Each candidate receives a separate Y/N query; its conditional Y likelihood is normalized over allowed values per field. Score s averages support for three verifier-selected values, used only after source hashes are saved. Reader input is task text, complete candidate support/opposing-evidence table (and their difference) after the fixed mixing transformation, and one candidate. It excludes event-source coordinates and raw evidence; it asks for best support, not an independently generated plan. Extraction uses temperature 00; scoring uses 1.01.0, top-p=1.0p=1.0. Seeds derived from 41014101 and candidate index match paired counterparts. This is not a calibrated joint-choice probability. Post-call exactness Q is not differentiated; workability is not varied. Source isolation and measurement units. All extraction blocks execute first. Each records normalized events, the graph and hashes of visible inputs, extractor requests/responses, events and graph, durably synchronized before the verifier file is opened. Historical records bind source version, served model identity and files, code, data, request/response identifiers and the planned denominator. Privately retained transformed copies replace identifying metadata and dependent hashes, checking internal consistency rather than authenticating original execution or chronology. All 44 confirmatory blocks complete without adaptive retry, repair or exclusion: 44 extraction plus 648648 scoring calls (652652 total). Endpoint calls comprise 432432 perturbation-arm and 216216 repeated-input/routing controls. Before verifier access, fixed ordering by event time, type, document, field, value and identifier selects three events per block from 1515 total: 1212 directions and 33 unprobed events. Calls are repeated candidate measurements within four tasks. Resolved contrasts and step refinement. A first-order interpretation requires a resolved readout slope that remains consistent when the perturbation is halved. This criterion is met in only some of the tested directions. Every selected direction exceeds the state-distance threshold by construction, with five distinct displacement magnitudes. Nine candidate slopes exceed threshold; the reactor-inspection instance’s three remain below it, not necessarily zero. Four of nine above-threshold slopes reverse sign, leaving five above-threshold, sign-stable directions. Only 2/122/12 meet fixed slope agreement: |g^ds(ϵ)−g^ds(ϵ/2)|max|g^ds(ϵ)|,|g^ds(ϵ/2)|,10−12≤0.25. | g^s_d(ε)- g^s_d(ε/2)| \| g^s_d(ε)|,| g^s_d(ε/2)|,10^-12\≤ 0.25. The floor normalizes near zero. Steps are 20%20\% and 10%10\% of baseline. Failure does not distinguish oversized steps from nonlinear or discrete behavior or establish zero population response. Post-run saturation counts are 547/648547/648 (84.4%84.4\%) scoring calls and 355/432355/432 (82.2%82.2\%) perturbation calls at maximum conditional Y/N probability at least 0.990.99; these counts do not isolate saturation’s contribution to insensitivity. Per instance, contrasts occupy distinct event coordinates in a 2121- or 2222-component state; their three-column finite-contrast matrix has zero kernel. This does not identify differential z,TK_z,T or test sufficiency. Two late coupling events and one support event remain unprobed. Repeated-input controls test aggregate-score repeatability. Each block’s two scores use nine calls each (72 of the 216 controls) and are identical; this establishes neither sampling independence nor identical raw responses. Restoration replaces both perturbed graph states with the baseline state; the no-propagation control disables edge transport. Each operation makes the paired candidate tables identical, and their readout differences are zero. These checks verify the numerical inputs, not causal mediation. Four variants of one template supply no task-family confidence interval, live multi-agent dynamics, natural unloaded reference or workability crossing. Within each fixed-task reader interface, access to original evidence is restricted to the projected support table. This design does not independently test the full graph’s sufficiency or its transfer across tasks. S7.2 Program-output diagnosis These checks use task-specific response relations to choose which computed output to replace. The retrospective examples and prospective pilot differ in when the diagnostic rule was selected relative to the observed failures. S7.2.1 Response checks on two saved programs Two saved model-written programs test whether a known input–output response can identify a faulty calculation. Each returns an integral and an inverse: time to a target volume for flow, or the coordinate at a target calibration value. Correction replaces a selected output with a reference formula while leaving the program unchanged. Programs and rules were selected after their defects were known; this is a retrospective diagnostic check. Task and diagnostic rule. At each of three positions, two sensor readings define a mean. The program receives a vector of prepared means and a separate worksheet of position-indexed values; both are represented in the graph. The separate input channels permit tests of which records the program uses. Flow rates (2,4,3)(2,4,3) litres/second over three one-second intervals give total volume 99 litres and time 13/813/8 seconds to 4.54.5 litres. Linear interpolation of calibration values (10,20,40)(10,20,40) at (0,1,2)(0,1,2) gives integral 4545 and target coordinate 1/21/2 for value 1515; these coordinates and values have no assigned physical units. A uniform shift ϵε of the readings must change an integral by ϵε times domain length. Each probe at ϵ=±0.1ε=± 0.1 adds ϵε to both sensor readings at every position, then recomputes their means and rebuilds both inputs while preserving any imposed additive preparation error. At tolerance 10−810^-8, checks compare preparation with sensor means, the inverse with the allowed domain, and the central-difference integral slope with domain length (3 for flow, 2 for calibration). A unique violation selects preparation recomputation, inverse replacement or integral replacement, respectively; multiple violations cause abstention, and execution/domain errors are unmeasurable. Reference endpoints and repair outcomes are excluded from selection. An incorrect inverse inside its domain can escape this rule. The flow program returns (9,25/6)(9,25/6), with target time outside the domain; inverse replacement gives (9,13/8)(9,13/8). The calibration program returns (15,1/2)(15,1/2); integral replacement gives (45,1/2)(45,1/2). Pairs denote (integral, inverse), in the units above. Unchanged execution, deterministic retry, identity intervention and replacement of the other output each leave both programs jointly incorrect; replacing both outputs corrects both. Nine constructed controls contain two healthy cases, two repaired preparation errors, two missed defects, one incomplete repair, one ambiguous case and one unsupported-domain case. Eight are scorable: selection gives 4/84/8 jointly correct outcomes versus 6/86/8 for both replacements. Healthy cases enter these counts, so they are not repair rates. Ordinary diagnostics have access to the same relations and formulas. Graph encoding and representation tests. The graph transports inputs, not program-generation dynamics. Six events write preparation/worksheet values as v/64v/64, v∈[0,64]v∈[0,64], into twelve sites and two channels. Positive polarity marks a write, not correctness. Forward/return weights 1,0.151,0.15, retention 1/21/2, propagation 1/41/4 and zero channel coupling produce six terminal action/channel values v/256v/256, decoded by multiplication by 256256; both updates are stored. Programs also receive raw sensor history, targets and durations/spacings. Blocking preparation fails domain checking; blocking worksheets leaves preparation readers unchanged but fails constructed worksheet readers. There is no fallback to original inputs. Restoring the original graph recovers the wrong answers. Substituting a perturbed graph while keeping the other inputs fixed reproduces the corresponding source-perturbation output. These controls establish the declared dependencies. The representation tests ask which of the supplied records are needed to account for program outputs. Identical worksheets with different preparation yield different outputs, rejecting worksheet-only state for those programs. A constructed reader that also uses raw sensors changes its total 1.8→2.01.8→ 2.0 at fixed graph, rejecting graph-only state for that wider input class. The six independent encoded input directions have zero state-response kernel and cannot provide positive sufficiency evidence. Readout choice is illustrated by rate direction (1,0,−1)(1,0,-1): after inverse replacement, V(δ)=9V(δ)=9 but t4.5(δ)=13/8−δ/4t_4.5(δ)=13/8-δ/4. A total-only readout misses this change. The two programs come from 30 development calls (6 preparations and 24 continuations), with no passes among 18 eligible accuracy evaluations. Reconstruction uses deterministic executions of these saved programs, with response checks at ±0.1,±0.05± 0.1,± 0.05, and no new model calls. Separate code checks graph updates, decoding, reference arithmetic, decisions and grading. The conclusions concern diagnostic logic and representation scope, not unknown-task repair performance or model-internal mechanisms. S7.2.2 Prospective checks on four new programs This pilot asks whether checks of preparation consistency, inverse-output range and integral response can select a correction before a new program’s defects are known. It uses two flow tasks (total volume and target time) and two calibration tasks (integral and target coordinate). The diagnostic rule, settings, actions and analysis were fixed and committed before Qwen2.5-14B generated one preparation and one program per case: eight fresh calls without retries or replacement cases. Corrective actions recompute preparation or replace selected outputs; generated code stays fixed. Inputs and outcomes. Each pair of readings equals its prescribed mean ±0.2± 0.2. Flow 01 uses means (6,2.5,6)(6,2.5,6), durations (3.5,2,3)(3.5,2,3) and target 1313; Flow 02 uses (5.5,7,3)(5.5,7,3), (2,3,2.5)(2,3,2.5) and target 3737. Units are litres/second, seconds and litres. Calibration 01 uses means (3,9.5,14)(3,9.5,14) and target 66; Calibration 02 uses (5,12.5,17.5)(5,12.5,17.5) and target 1515. Both interpolate linearly at (0,1,2)(0,1,2), with no physical units. Reference integration and inversion use the preceding task definitions. Decisions precede outcomes; 64 deterministic graph executions compare all actions and controls. Table S5 shows which outputs each action corrects. Table S5: A corrected component need not repair the task. Each of four newly generated programs returns two outputs. Flow outputs are total volume and time to a target volume; calibration outputs are the integral of an interpolated curve and the coordinate at a target value. Counts compare correctness before and after the selected action; any correction replaces numerical outputs without changing the program. A task is correct only at 2/2. Both healthy programs remain correct, while neither faulty program is fully repaired. Exact outputs and reference calculations are retained in the authors’ records. Program Correct outputs Action and remaining error Flow 01 1/2→1/21/2→ 1/2 No change; target time remains incorrect Flow 02 2/2→2/22/2→ 2/2 No change; both outputs correct Calibration 01 2/2→2/22/2→ 2/2 No change; both outputs correct Calibration 02 0/2→1/20/2→ 1/2 Integral replaced; target coordinate remains incorrect Interpretation. The selected integral correction leaves both faulty programs’ inverse outputs incorrect (Table S5). Calibration 02 uses (p(1)−p(0)+p(2)−p(1))/2(p(1)-p(0)+p(2)-p(1))/2 for decoded preparation values p: a uniform shift cancels, exposing the wrong integral response. Replacement gives the reference 95/495/4. Its inverse, 4/34/3 rather than 3/23/2, nevertheless passes the domain check [0,2][0,2]; Flow 01’s wrong time 6 rather than 13/613/6 likewise lies in [0,8.5][0,8.5]. Response and range checks can identify one defective output while missing another. Joint correctness stays 2/42/4, component correctness changes 5/8→6/85/8→ 6/8, and full repair is 0/20/2; neither healthy case changes. Across all four cases, unchanged execution, deterministic repeat, identity intervention, preparation/worksheet rebuilding and integral replacement each give 2/42/4 joint correctness; inverse replacement gives 3/43/4, and both reference replacements 4/44/4. The latter is an arithmetic control, not model self-repair; no ordinary diagnostic comparator is implemented. Graph transplants change at least one readout per case; preparation blocking is inadmissible, worksheet blocking leaves Calibration 02 unchanged, and restoration returns wrong outputs. No case meets the fixed criterion of full repair together with a resolved pre-intervention response of the diagnosed readout. A separate calculation checks recorded decisions, executions and references; these four cases provide no comparative diagnostic advantage. S7.3 Presentation changes in a delayed-tool workflow An inventory reader associates delayed quantity returns with requests, then decides whether stock suffices for an order. The experiment tests whether explicitly pairing existing input records restores a missing response. Four constructed cases cover irrelevant-warehouse returns, reversed arrival/revision order, cancellation and repeated delivery (cases 1–4). Delays are prescribed event order, not measured latency. Task and controlled contrast. Qwen2.5-14B-Instruct-MLX-4bit receives all requests (identifier, warehouse, revision and active/cancelled status), identified returns, a snapshot and an unrelated note. It should select the highest active revision for the target warehouse up to the current revision, then allocate only if its quantity suffices. A fixed warehouse–revision lookup supplies quantities. Exchanging two requests’ warehouses preserves their returns and arrival order. Snapshot state zPz_P records target warehouse, order quantity and current revision; zHz_H additionally records requests. The exchange preserves zPz_P but changes zHz_H. These analytical descriptions do not restrict model access to the full input. The cancellation case (case 3) shows why arrival order alone is insufficient. It requests ten units at revision seven. Three requests, A, B and C, return in order B, A, C: A is active at revision six for the target warehouse and returns 14 units; B is active at revision five for the other warehouse and returns 9; C is cancelled at revision seven for the target and returns 25. Exchanging A’s and B’s warehouses changes the selected quantity 14→914→ 9, and the required action from allocate to withhold. This is the −5-5-unit response in Figure S2. Calibration and exploratory interventions. Calibration reproduces the required quantity changes in cases 1, 2 and 4, while case 3 stays at 9→99→ 9. Four cases, two conditions and three repeats give 24 calls: all exact-key JSON outputs parse, but only 21/24 give the correct quantity and action, below the fixed 22/24 criterion. The planned 320-call exploration/validation therefore remains unrun. The task list, response rule and direct association diagnostic were fixed before calls, without independent task custody. Settings are temperature 0.2, top-p 1, 128-token output cap, 8192-token context and 90-second deadline, with fixed case-derived condition shuffling and no repair or retry. Both response-based and direct association checks select pairing for case 3 and no change elsewhere before intervention outcomes. Pairing appends every request joined to its return by identifier; it does not select an answer. A separately specified exploratory comparison runs six operations on all four cases, three times each (72 fresh calls). Join, join plus checklist, and an idealized refresh each give 12/12 jointly correct quantity/action outputs; unchanged retry, an unrelated note, and checklist alone each give 9/12, as do the twelve unperturbed calibration calls (excluding the twelve source-contrast calls). All differences occur in case 3 (3/3 versus 0/3); the other three cases contribute 9/9 in every operation. The checklist restates filtering, revision choice and allocation. Refresh supplies an authoritative quantity and is answer-informed. No failed call is replaced. A descriptive cost comparison also favors a fixed join. Utility assigns +1+1 to jointly correct output, −2-2 to unsafe allocation, −1-1 to invalid output and zero otherwise, minus 0.01 per deployment call and 0.05 per refresh. Both diagnostic policies use six probes per case and one selected continuation on case 3, reusing historical outputs elsewhere. Their mean utility is 0.9375 versus 0.99 for fixed join or join plus checklist; unused-arm collection is excluded. These illustrative weights are not measured costs, and fixed strategies have no independent selection set. Ordinary diagnosis selects the same action; the deterministic task rule could also be executed directly. Same-case response restoration. A separate exploratory test repeats both conditions of the known failure case: four operations, two inputs and three repeats give 24 fresh calls. The protocol, randomized schedule and reconstruction were fixed before these calls; the case and successful join were already known. No historical output enters a pair, and pair indices do not imply shared seeds. All outputs parse. The criterion requires both correct endpoints and the exact signed difference in all three join pairs; Figure S2 reports every result. Figure S2: Pairing existing records restores a missing response on one known case. The task selects the highest active request revision for the target warehouse, excluding revisions later than the current one, and allocates only when the returned quantity meets the order. In case 3, the order requires ten units and the current revision is seven. The source contrast exchanges two requests’ warehouse assignments while keeping returned quantities fixed. a, Required changes (squares) and observed changes (circles) in four cases. Only the cancellation case (case 3) lacks the required response: zero rather than −5-5 units. Small vertical offsets separate coincident markers. b, A separate 24-call follow-up on case 3. Explicitly pairing every request with its return by identifier gives 14/allocate→9/hold14/allocate→ 9/hold: all three pairs have correct endpoints and difference −5-5. Unchanged input, a checklist, and an unrelated note each give 9/hold→9/hold9/hold→ 9/hold, or 0/3 correct pairs, and share the dashed line. Observed values summarize three repeated calls per condition or three matched response pairs, not independent tasks; squares in panel a show task-rule reference contrasts. Lines in panel b connect two inputs, not times. The pairing changes presentation, not available information; the test covers the same selected case rather than held-out repair performance. Restoring the two-point response rules out an always-14 answer; the unchanged 9/hold9/hold endpoint alone cannot do so. Both the full task rule and choosing the first target-warehouse return remain compatible with the result. Joining changes presentation, salience and length, rather than available information. This same-case exploratory result identifies neither a unique mechanism nor state sufficiency, held-out repair performance or diagnostic advantage. The note control supplies no distributional-equivalence bound. Exact requests and responses remain in the authors’ records. S7.4 Development study of response-guided action selection This development study asks whether measuring changes in selected evidence helps choose an intervention that improves a claim-verification answer. It compares selection from state and endpoint responses with endpoint-only features and direct reading of the retrieved evidence. SciFact claims 57 enter a two-stage pipeline: evidence selection followed by claim classification. BM25 keyword ranking retrieves six abstracts from the claim text alone. Qwen2.5-14B-Instruct (MLX 4-bit) selects sentence identifiers; a source-checked parser resolves these to ordered original text. A second call reads only the claim and selected evidence, returning support, contradiction or insufficient evidence. Reference labels score correctness. Format conversions preserve source text; invalid selections, unlike valid empty ones, skip the reader and leave its outcome unavailable. Serial calls use fresh caches, explicit seeds, temperature 0.7, no probability truncation or repetition penalties, and output limits of 512 tokens for evidence selection and 16 for claim classification. Fixed hash order selects one training claim per source group linked by citations or duplicates, excluding earlier design and pilot groups. Of 96 tasks, 89 complete, one is partial and six unrun. A post-interruption amendment, made before scoring, includes the completed initial segment without imputation. Each task has two recorded sets of action outcomes, sharing some calls: 89 groups, not 178 independent tasks. Outcomes average the two sets within each task. Actions and comparison methods. Five actions keep the original answer, reselect evidence, change the reader prompt at fixed evidence, read all six abstracts, or aggregate upstream and altered-reader answers. Aggregation returns their common valid label or uses another call with visible evidence and answers, never reference labels. Consistency voting uses up to eight readings of the original evidence. All actions are evaluated on the original, undeleted task. The feature comparison asks whether observing a response helps beyond ordinary answer and evidence diagnostics. Two probes delete the first-ranked abstract or first three, without replacement. Ordinary features B include validity, label, evidence length and retrieval rank, retrieval scores, available confidence, missingness and changes caused directly by deletion. Endpoint features EiE_i add label/score contrasts and repeat controls. State features SiS_i also measure whether the selector changes its choice among surviving sentences; matched E/SE/S sets share all other values and availability indicators. The index i counts probes. One probe uses the first deletion and fixed-evidence rereading; two add the second deletion and repeated selection and classification, with maximum diagnostic costs of five and nine calls. Consistency features K add repeated-answer statistics to B. The primary comparisons are S2S_2 against E2E_2 and direct reading; S1S_1 comparisons are secondary. Actions maximize predicted repair minus harm relative to the original answer, with fixed ties. Six ridge predictors use five fixed group folds, training-fold median imputation/standardization and penalties 0.1,1,10,100,1000\0.1,1,10,100,1000\ chosen by Brier loss (squared probability error). Six sensitivity comparators use single-split probability rules, including a constant candidate, with training-only features, thresholds and leaf frequencies. Tuning reuses assessment folds, and accuracy-first fixed-action selection uses the same development outcomes. These are descriptive comparisons, not independent confirmation. Stored chronology does not independently authenticate label withholding or original request provenance. Figure S3: The primary response-guided strategy matches direct reading’s observed correctness at higher call cost. a, Correct-label fractions for all declared methods on 89 completed SciFact tasks. b, Mean calls from task start, including diagnostic measurements and the selected action; shared calls count once. Both panels average two sets of action outcomes within each task and then across tasks: 89 source groups, not 178 independent samples. Invalid or unavailable labels remain incorrect. Highlighted rows are direct reading and the primary state-response strategy S2S_2. Both have 65/89 and 68/89 correct outcomes in the two recorded sets, averaging 74.72%, at 1.00 and 7.56 calls per task. B uses answer and evidence diagnostics, including validity, labels, retrieval information and changes caused directly by deletion. The index i=1,2i=1,2 counts probes: remove the first-ranked abstract, then additionally test removal of the first three. EiE_i adds changes in the reader’s label and score; SiS_i also measures changes in the selection of surviving sentences beyond mechanical deletion. K adds repeated-answer statistics. Direct reading supplies all six abstracts. Identical simple-rule pairs share a row. These are development estimates: tuning and fixed-action selection reuse the assessment data. Point differences do not establish superiority or equivalence. Per-method counts and token costs remain in the authors’ records. Results and measurement coverage. The primary response-guided strategy S2S_2 matches direct reading and E2E_2 in observed correctness on all 178 recorded task outcomes (Figure S3). It costs 7.56 calls per task, compared with 1.00 for direct reading. The ordinary-feature selector B, without state or endpoint response features, gives 66/89 and 68/89 correct answers in the two outcome sets (75.28% on average), at 2.55 calls per task. Secondary S1S_1 exceeds E1E_1 and direct reading by 1.12 percentage points: one task is corrected in both outcome sets by changing the reader prompt. This does not replace the primary comparison. These development estimates establish neither population superiority nor equivalence. The response measurements require a usable evidence selection. Initial selection calls produce 44 valid, 42 invalid and three capped/unavailable outputs. The invalid outputs comprise 24 sentence-object schema errors, eight failures to reproduce source text, six top-level key errors and four JSON parsing errors; sentence objects occur despite requests for identifiers. Reselection among surviving sentences is measurable on 23/89 tasks after removing the first-ranked abstract and 28/89 after removing the first three: 16 allow both measurements, 35 at least one, and 54 neither. Seventeen calls reach the output cap, including the three unavailable initial evidence selections. All 89 tasks stay in the denominator. Missing measurements are not zeroes and do not establish the cause of absent primary benefit. The accuracy contrast between selected-evidence and direct-reading paths does not compare matched states; it cannot reject a representation. The stopped run has 1,511 attempts, one with unknown completion; independent confirmation was not performed. Recorded outputs permit reanalysis of policy choices and observed costs, without authenticating historical execution. Costs of the unknown completion remain unavailable. S7.5 Build-and-test workflow: task and screening results An authored software workflow asked whether required packages could be released: every prerequisite had to use the correct component and source revision, remain uncancelled, and return successfully within the observation window. Build and test operations were executed locally; delivery times were assigned. Candidate summaries are tested by comparing release, hold and invalid-output frequencies for histories they merge. Correctness is scored separately against the prerequisite rules; agreement can include a shared incorrect judgment. Qwen2.5-14B-Instruct (MLX 4-bit) read complete histories and tool returns; candidate states and reference judgments were withheld from its inputs. Unlike the closed-loop task (Section S5), the model reports only the release judgment, without first reporting current-version test statuses. The task rules, record presentation and output requirements also differ; this is not a matched interface ablation. Each history contains build, test and package jobs for two components. Four fault families alter a prerequisite’s component, its source revision, a required test’s cancellation status, or the build revision used by a test. Each family has healthy and faulty variants, each with two preparations that differ in an unrelated job, note and dispatch time. Five candidate representations retain: (i) current revisions and the pending-job count; (i) this snapshot, required package jobs, and each job’s component and operation; (i) the preceding description plus job revisions and cancellation status; (iv) current revisions and the dependency graph of required package jobs and their transitive prerequisites; or (v) the full history. The graph consistently renames identifiers and omits jobs outside those dependencies, while the model still sees the original complete histories. This candidate library is not a nested sequence. The readout distinguishes a release decision, a hold decision and an invalid response. READY and HOLD denote valid JSON judgments with the single field ready set to true or false, respectively; all other complete responses are INVALID. Sixteen ten-job histories, eight healthy and eight faulty, received four calls each: 64 discovery calls plus eight preliminary sampling checks. Discovery requests temperature 0.70.7, top-p 11, top-k 00, distinct fixed seeds and a 64-token output limit. Eight unscored preliminary calls vary omitted, distinct or repeated seeds and zero temperature, while allowing either Boolean answer. All return READY, leaving sampling independence and effective seed/temperature handling unverified. Faulty prerequisites often leave the release judgment unchanged: strict correctness is 32/64, the count obtained by always answering READY. All 32 calls on healthy histories returned READY, as did all 24 calls across six faulty histories with wrong-component prerequisites, obsolete source revisions or required tests using obsolete builds. The two cancellation histories gave one READY and seven INVALID responses. The invalid responses began with a false judgment but added prose and reached the output limit; their classification is unchanged. These 32 calls on eight faulty histories are repeated measurements, not independent tasks. No candidate met both requirements for response agreement and comparison coverage. Selection compared distinct histories assigned the same candidate state within each fault family, requiring at least one pair per fault family and empirical total variation—the largest probability difference over output events—at most 0.150.15 on every pair. The graph retains component associations, revisions, cancellation statuses and dependency edges; it had eight matched pairs within the prescribed families. One cancellation pair gave empirical distance 0.250.25, the other seven zero. With four draws per history, 0.250.25 is the smallest nonzero histogram distance. Snapshot and association-only candidates also exceeded the histogram threshold. The revision/status description had matched pairs in only one family, while full history had none; both lacked the required comparison coverage. The fixed stopping rule left all 1,024 confirmation calls unrun. This preliminary comparison is not a formal statistical rejection of the graph representation. The observations do not isolate the causes of task errors or format failures and provide no positive confirmation of representation adequacy. S7.6 Historical readout comparisons These historical records compare tool use with answer correctness as the permitted number of tool steps decreases. Tool use defines PfP_f; tool use with a correct answer defines the nested readout QfQ_f. Write w=Pfw=P_f and r=Qf/Pfr=Q_f/P_f for Pf>0P_f>0. The comparisons illustrate these fractions, with retrospectively selected paths rather than a common response law. Spreadsheet paths and nested readouts. Eight historical paths vary tool-step budgets on author-constructed spreadsheet tasks derived from GAIA 32. They are not official benchmark evaluations. Seven paths, selected retrospectively across three spreadsheet task groups, decrease from exact or partially exact operation to failure. None visits all four regimes. The eighth path, on another spreadsheet, has (w,r):(1,5/6)→(1,1)→(0,−)(w,r):(1,5/6)→(1,1)→(0,-), contradicting monotonicity. Three workflow implementations provide 12 records on four tasks and 36 on twelve tasks. At tool budgets 2 and 1, respectively, all are workable; exact outcomes are 12 and 10 in the smaller set, 25 and 22 in the larger. Thus ratios such as 11/18=22/3611/18=22/36 describe pooled outcomes, not an 18-record sample. At zero budget w=0w=0 and r is undefined. Separate constraint-planning, recommendation and delayed-counting records illustrate workable but inexact outcomes. Their different controls and readouts, incomplete counts and retrospective selection do not support a common response law. The ordered boundary result requires Section S2.4.1’s hypotheses. Records contain task identifiers, answers, correctness, tool use, tool-step budgets, run indices, elapsed time and model-call counts. They lack the stopping rule, independent pre-scoring timestamp, temperature, seed, model revision and per-run tool/verification counts. They therefore support coordinate illustrations, not complete execution reproduction or pre-scoring protocol attestation. These paths are distinct from the seven live-origin records used for graph replay, although four official task identifiers overlap. S7.6.1 Exploratory persistence records A historical rule-based counting task tests information carried between steps by removing, masking or restoring it while fixing task, model family, interface, actions and stopping rule. It also records a persistent reference, a mixture-of-agents baseline using aggregated summaries, and a single-agent snapshot. No cross-agent propagation was measured. Removal gives a wrong answer; the original baseline, masking and restoration time out. A separately recorded persistent reference and the mixture baseline answer correctly. Restoration recovers information-visibility proxies but not the answer. Removal, masking, mixture and snapshot each fail the joint criterion of no worse recovery and no greater resource use than the persistent reference, with at least one strict improvement. These proxies lack complete formulas and denominators; they are neither probabilities, nested (Pf,Qf)(P_f,Q_f) readouts nor the coherence functional of Section S2.3. The removal, masking and restoration reruns duplicate earlier records byte-for-byte. Missing execution settings, an unreleased fuller report and no snapshot recovery trajectory preclude source-level reconstruction, independent replication or a general necessity claim. Derived comparisons remain in the authors’ records. S8 Evidence status, data and code availability S8.1 Evidence status and reconstruction limits Analytic results, comparisons fixed before measurement, and developmental or exploratory analyses support different inferences (Table S6). Each study’s methods specify its task, comparison rule and sampling assumptions. Local hashes and timestamps bind files; they do not independently authenticate registration or historical execution. Without raw records and implementations, readers cannot numerically reanalyze all reported results. Table S6: Scientific role and inferential scope of the evidence. New-history comparisons are distinct from independent tasks or external replication. Sampling-based claims remain conditional on each study’s stated assumptions. Analyses of the same observations are not independent evidence. Study Evidence status Supported interpretation and limit Physical measurements Analytic response and numerical calibration Local consensus calibration and distinct flock readouts; neither empirically tests inverse state adequacy. Request–return matching Constructed program and proof Association suffices for the specified continuations; executions check its implementation. Recipient-A queries Readout selected after development; comparisons fixed before new calls Bounded support on four histories, restricted to recipient A. Varied identifiers and order Fixed comparisons; exploratory reanalysis Three pairs remain unresolved under original bounds; post-hoc categorical bounds resolve differences. Added identifiers and order Fixed new comparisons Association and extended states both have bounded support; added benefit is unestablished and the earlier anomaly was not retested. Adaptive tests Instructed task; fixed confirmation after interface development Feedback tests prescribed version information; the instructed rule predicts every observed action. Excerpt comparison Fixed comparisons on eight SciFact training claims Categorical bounds reject both compressed descriptions; neither is adequate across the claim set. Build and test Stopped development screen No candidate selected or confirmation run; not an interface ablation of the adaptive task. Delayed tools and programs Constructed cases and pilots Specific presentation changes and output replacements; no independent decision benefit established. Graph readout Imposed propagation; fresh model readouts Finite differences test local linearity; each graph’s three-direction state-response matrix has full column rank. Development decisions Reused assessment folds Primary selection matches direct reading at greater cost; missing measurements do not explain this result. Historical persistence Exploratory records Incomplete provenance and scoring limit authentication and interpretation. S8.2 Data availability The physical studies use numerical simulations of delayed consensus and polar flocking. AI data comprise constructed event graphs and request–return histories, saved model-generated programs, and recorded model responses. The excerpt-comparison and action-selection studies use SciFact’s training set and abstracts 57, available at https://github.com/allenai/scifact. Historical comparisons use author-constructed variants derived from GAIA 32, rather than an official benchmark evaluation. The manuscript and SI describe the task inputs, sampling and scoring rules, controls, and aggregate results, including unsuccessful comparisons. Raw experimental records and physical simulation arrays have not been publicly deposited. Consequently, the publication alone does not permit reconstruction of every reported figure or response comparison. The source dataset’s public availability does not supply the constructed prompts, generated responses or simulation records. S8.3 Code availability The manuscript and SI specify equations, task rules and statistical procedures for independent implementation. LaTeX sources reproduce submitted documents and figures but provide no executable experimental reproduction package. Experiment, analysis and plotting implementations are not publicly released. S9 Notation and terminology A loading parameter is a scalar or vector control varied about a fixed reference protocol. A finite-window response maps a source perturbation to retained-state and endpoint-readout changes during a fixed window. Each graph experiment specifies its own coordinates and event-to-coordinate map. The model-readout graph separates event sources, candidate support and opposing evidence; the program graph encodes preparation and worksheet values. These externally constructed states are neither graph neural networks nor model hidden states. Prescribed delayed inputs originate outside the modeled local operators; delay alone does not imply endogenous feedback. Internally generated delayed quantities require their histories in the augmented state (retarded dynamics). Retarded response specifies causality—zero response before the source acts—independently of endogenous delayed feedback. An operating branch is a local regular branch of retained-state equilibria parametrized by control; stability is a separate condition. A slow manifold is a locally invariant manifold on which eliminated variables are slaved to retained coordinates, where such a reduction exists. A linear-system residual is the back-substitution error for a linear solve using assembled A,BA,B; it does not check matrix assembly. Table S7: Selected symbols. Full-state, retained-state and readout roles are distinguished; realization-specific meanings are identified explicitly. Symbol Meaning and scope ct∈Hc_t∈ H shared coordination field in Hilbert field space H, the field component of the augmented state st∈Xfs_t∈ X_f full augmented state containing the field, memories, required delay histories, direct readout inputs and observation records zT=Gf(sT)∈Zfz_T=G_f(s_T)∈ Z_f candidate retained state; may include the full field. Low dimension, endpoint sufficiency and autonomous closure each require separate justification OT:Xf→fO_T:X_f _f endpoint readout on the full augmented state, with observation space fY_f; numerical observables take values in ℝdR^d O¯T:Zf→f O_T:Z_f _f readout as a function of retained state, where OT(s)=O¯T(Gf(s))O_T(s)= O_T(G_f(s)); the bar distinguishes the domain and does not denote an average DxF(x)D_xF(x) derivative at x, mapping an input perturbation to a first-order output change; a Jacobian in finite dimensions or a Fréchet derivative on a function space DsOTD_sO_T readout sensitivity to the full state; under differentiable factorization it equals DzO¯TDsGfD_z O_T\,D_sG_f, evaluated at the corresponding state values O¯fi,T O_f_i,T, ireachZ_i^reach realization-specific projected readout and reached retained-state set in a two-realization comparison d≥1d≥ 1, q readout-component count and field-channel count; neither fixes augmented- or retained-state dimension utu_t, δutδ u_t, Δut u_t controlled source coordinates, tangent perturbation and finite source contrast; u also parametrizes a fixed operating branch TU_T declared finite-dimensional source-perturbation space over the observation window z,T,O,TK_z,T,K_O,T first-order maps from TU_T to retained-state and endpoint displacements ℋT(1)H_T^(1) first-order readout map on reached state displacements; exists when every source direction with zero state response also has zero readout response cons=(,ϕ,V) y_ cons=(A,φ,V) consensus window summary: amplitude, phase lag and transverse variance, evaluated on the observed field trajectory σθ(P) _θ^(P) flock width inferred from PfP_f; a steady-state consistency summary rPRr_ PR participation ratio using squared singular-value weights of the constructed response matrix reffr_ eff consensus participation ratio using E0=2E_0=A^2 and transverse weights summing to V ht,at,rt+1h_t,a_t,r_t+1 (adaptive workflow) visible interaction history, model-selected action and subsequent delivered tool observation π(a∣h),qu(r∣h,a)π(a h),q_u(r h,a) fixed decision policy and tool-observation law under external condition u; neither implies autonomous evolution of a candidate summary At,Bt,Φ(T,t)A_t,B_t, (T,t) linearized full-state transition, source Jacobian and ordered transition product Atret,Btret,Φret(T,t)A_t ret,B_t ret, _ ret(T,t) corresponding maps for a closed retained-state linearization, where that closure holds A,BA,B, ρ(A)ρ(A) time-independent transition and source maps of the closed retained graph; spectral radius of A α,βα,β (event graph) dimensionless uniform retention and transport gains; fixed settings 0.72,0.450.72,0.45 L, JiJ_i intrinsic field operator and composite local source Ri,πi,WiR_i, _i,W_i local read, update and write maps GiG_i source-coupling operator for a modeled local operator EℓE_ , τℓ,yℓ _ ,y_ coupling, delay and prescribed signal for a delayed input whose generator remains outside the modeled local operators ξat _a^t history of an internally generated delayed quantity, retained in s(t)s(t) C(t)C(t) coherence functional, Section S2.3 Ffull(s,u,yt)F_ full(s;u,y^t) augmented controlled evolution, including memory laws and internal history transport; yty^t is prescribed fu(z)f_u(z) reduced drift in z˙=fu(z) z=f_u(z), where the stated local reduction exists Pf,QfP_f,Q_f (nested criteria) outer workability and inner exactness; logical inclusion and normalization give 0≤Qf≤Pf≤10≤ Q_f≤ P_f≤ 1. Numerical order alone does not establish nesting P^f,Q^f P_f, Q_f (consensus) sample support fractions obtained by averaging the binary readouts over the same 5050 seeds Pf,QfP_f,Q_f (flock) time- and seed-averaged polarization and alignment fraction; these observables are not nested w=Pfw=P_f, r=Qf/Pfr=Q_f/P_f workability and inner-to-outer ratio for nested criteria; w remains defined at Pf=0P_f=0, where r is undefined r=Qf/Pfr=Q_f/P_f (flock) alignment fraction divided by polarization for Pf>0P_f>0; not conditional exactness; the nested-readout interpretation of (w,r)(w,r) does not apply Pfprep,GfP_f prep,G_f initial-state preparation and full-to-retained-state coarse-graining maps; preparation differs from subsequent dynamics and scalar PfP_f χ^G(t,t′) χ^G(t,t ) finite-graph finite-window response χ∗G=(I−A)−1B _ ^G=(I-A)^-1B time-summed graph response for time-independent stable dynamics χ∗ _ continuous-time stationary branch susceptibility −Az−1Bz-A_z^-1B_z, under the stated branch hypotheses ΣEW,ΣWR,ΣRO _EW, _WR, _RO regular readout boundaries; the stated sequence requires actual crossings, no re-entry and ordered, separated thresholds on a fixed path Table S7: Selected symbols (continued). 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